Group action - Knowledge

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of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a

6906:. The definition can be made more precise by specifying what is meant by image or pattern, for example, a function of position with values in a set of colors. Isometries are in fact one example of affine group (action). 2436:

in a symmetric group (which is infinite when the group is). A finite group may act faithfully on a set of size much smaller than its cardinality (however such an action cannot be free). For instance the abelian 2-group

6098: 8118: 2256:

Thus, for establishing general properties of group actions, it suffices to consider only left actions. However, there are cases where this is not possible. For example, the multiplication of a group

5195: 8106: 2007: 1276: 2160: 1746: 3947: 4767: 1873: 1138: 5622: 2480:

In general the smallest set on which a faithful action can be defined can vary greatly for groups of the same size. For example, three groups of size 120 are the symmetric group

1435: 1927: 1196: 2089: 1680: 504: 479: 442: 5977:

fixing 1, 2 and 3 must also fix all other vertices, since they are determined by their adjacency to 1, 2 and 3. Combining the preceding calculations, we can now obtain

1446: 806: 1038:

of a set depends formally on the set, the concept of group action allows one to consider a single group for studying the permutations of all sets with the same

8001:

has an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtain

7625:

associated to the group action. The stabilizers of the action are the vertex groups of the groupoid and the orbits of the action are its components.

5934:

that fixes 1 and 2 must send 3 to either 3 or 6. Reflecting the cube at the plane through 1, 2, 7 and 8 is such an automorphism sending 3 to 6, thus

5854:

that fixes 1 must send 2 to either 2, 4, or 5. As an example of such automorphisms consider the rotation around the diagonal axis through 1 and 7 by

8812: 6001: 364: 8732: 8667: 8617: 8598: 8548: 8524: 8478: 2232:, a left action can be constructed from a right action by composing with the inverse operation of the group. Also, a right action of a group 8807: 5766:, and this action is transitive as can be seen by composing rotations about the center of the cube. Thus, by the orbit-stabilizer theorem, 6124:

are finite, when it can be interpreted as follows: the number of orbits is equal to the average number of points fixed per group element.

5514: 314: 6591:

of a polyhedron acts on the set of vertices of that polyhedron. It also acts on the set of faces or the set of edges of the polyhedron.

3889: 8368: 799: 309: 8579: 8505: 8410: 8323: 4704: 7973:

Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary category: start with an object

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on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. See

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If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of

8199: 8181: 5739: 8761: 3776: 3065: 2974: 2634: 1572:. The second axiom then states that the function composition is compatible with the group multiplication; they form a 409: 223: 8756: 2512:. The smallest sets on which faithful actions can be defined for these groups are of size 5, 7, and 16 respectively. 2260:

both a left action and a right action on the group itself—multiplication on the left and on the right, respectively.

4792: 4424: 4060: 3744: 1837: 1102: 8677: 8039: 4813: 4334: 3390: 2839: 1350: 846: 6697:, or group, or ring ...) acts on the vector space (or set of vertices of the graph, or group, or ring ...). 5717:

This result is especially useful since it can be employed for counting arguments (typically in situations where

5355:{\displaystyle f(g)=f(h)\iff g{\cdot }x=h{\cdot }x\iff g^{-1}h{\cdot }x=x\iff g^{-1}h\in G_{x}\iff h\in gG_{x}.} 6555: 607: 341: 218: 106: 28: 8145: 8060: 6872:. This is a quotient of the action of the general linear group on projective space. Particularly notable is 6844: 6808: 3676: 3411: 3016: 966:

of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with

8049: 6916: 3361: 2257: 1094: 871: 858: 834: 819: 757: 547: 8802: 6687: 3351: 2428:

For example, the action of any group on itself by left multiplication is free. This observation implies

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Orbit of a fundamental spherical triangle (marked in red) under action of the full icosahedral group.

8088: 8002: 7381: 6832:, and the subgroup V of the affine group (that is, a vector space) has transitive and free (that is, 5631: 5116: 3980: 3736: 3724: 2948: 2931: 1569: 971: 963: 911: 854: 571: 559: 177: 111: 8638:, Grundlehren der Mathematischen Wissenschaften, vol. 287, Springer-Verlag, pp. XIII+326, 8112:

Orbit of a fundamental spherical triangle (marked in red) under action of the full octahedral group.

1891: 1160: 8719:, de Gruyter Studies in Mathematics, vol. 8, Berlin: Walter de Gruyter & Co., p. 29, 8080: 7085: 6780:. The group operations are given by multiplying the matrices from the groups with the vectors from 6288: 5995: 5411: 3271: 2429: 1573: 1060: 999: 986: 850: 146: 41: 2060: 1651: 487: 462: 425: 8466: 7943: 7581: 6396: 6291:– that every group is isomorphic to a subgroup of the symmetric group of permutations of the set 5753: 3970: 2996: 2838:(that is, subgroups of a finite symmetric group whose action is 2-transitive) and more generally 2835: 924: 901: 131: 103: 3656:. Contrary to what the name suggests, this is a weaker property than continuity of the action. 8817: 8775: 8728: 8663: 8613: 8594: 8575: 8544: 8520: 8501: 8474: 8406: 8364: 8319: 8076: 7089: 6225: 6154: 4023: 3786: 3771: 3423: 3275: 3034: 2908: 1025: 944: 892: 702: 536: 379: 273: 8215: 1554:{\displaystyle \alpha _{g}(\alpha _{h}(x))=(\alpha _{g}\circ \alpha _{h})(x)=\alpha _{gh}(x)} 8720: 8702: 8655: 8639: 8623: 8064: 8029: 7967: 7093: 6859: 6756: 6729: 6598: 6367: 4550: 4209: 3740: 3712: 3451: 2958: 1064: 687: 679: 671: 663: 655: 643: 583: 523: 513: 355: 297: 172: 141: 8742: 8701:, Princeton Mathematical Series, vol. 35, Princeton University Press, pp. x+311, 8558: 8488: 8738: 8706: 8643: 8627: 8554: 8484: 8072: 8015: 7939: 7744: 7370: 7002: 6903: 5047: 5043: 4566: 3755: 3282:

is wandering and free. Such actions can be characterized by the following property: every

1011: 952: 948: 928: 771: 764: 750: 707: 595: 518: 348: 262: 202: 82: 6885: 3804:(of order 60/12 = 5) is naturally identified with the 5 tetrahedra – the coset 3389:. For a properly discontinuous action, cocompactness is equivalent to compactness of the 2947:

of non-zero vectors is transitive, but not 2-transitive (similarly for the action of the

6858:

and its subgroups, particularly its Lie subgroups, which are Lie groups that act on the

6287:. This action is free and transitive (regular), and forms the basis of a rapid proof of 3337:, and the largest subset on which the action is freely discontinuous is then called the 3156:

of the action is the set of all points of discontinuity. Equivalently it is the largest

6880:, the symmetries of the projective line, which is sharply 3-transitive, preserving the 6588: 6150: 4213: 3993: 3554: 3279: 2241: 862: 778: 714: 404: 384: 321: 286: 207: 197: 182: 167: 121: 98: 6594:

The symmetry group of any geometrical object acts on the set of points of that object.

8796: 8538: 8534: 8150: 7986: 7366: 6140: 5743: 5446:

is contained in such a coset, and every such coset also occurs as a fiber. Therefore

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of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational)

3193: 3181: 3044: 697: 619: 453: 326: 192: 5634:. In particular that implies that the orbit length is a divisor of the group order. 8092: 8084: 7081: 6998: 6837: 6829: 6821: 6713: 6694: 6529: 2461: 2336: 959: 824: 552: 251: 240: 187: 162: 157: 116: 50: 8056:

is a functor from the groupoid to the category of sets or to some other category.

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if and only if all elements are equivalent, meaning that there is only one orbit.

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a topological space on which it acts by homeomorphisms. The action is said to be

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The difference between left and right actions is in the order in which a product

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The defining properties of a group guarantee that the set of orbits of (points

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acts also on various related structures; for example, the above rotation group

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and also on the figures drawn in it; in particular, it acts on the set of all

719: 447: 6836:) action on these points; indeed this can be used to give a definition of an 2834:

this is often called double, respectively triple, transitivity. The class of

1614:

entirely, and to replace it with either a dot, or with nothing at all. Thus,

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around a point in the plane. It is often useful to consider the group as an

540: 27:

This article is about the mathematical concept. For the sociology term, see

7570:-set has the property that its fixed points correspond to equivariant maps 7166:

acts on the set of real functions of a real variable in various ways, with

3011:

apart from the trivial partitions (the partition in a single piece and its

2961:

of a Euclidean space is not transitive on nonzero vectors but it is on the

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by natural matrix action. The orbits of its action are classified by the

6425:. An exponential notation is commonly used for the right-action variant: 4770: 4535: 1340: 1034:

elements by permuting the elements of the set. Although the group of all

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We can use the orbit-stabilizer theorem to count the automorphisms of a

1645:, especially when the action is clear from context. The axioms are then 8019: 7922:

can be taken to be the stabilizer group of any element of the original

4570: 3231:. This is strictly stronger than wandering; for instance the action of 419: 333: 8683: 8363:(1st ed.). The Mathematical Association of America. p. 200. 8063:

of topological groups on topological spaces, one also often considers

5042:

The above says that the stabilizers of elements in the same orbit are

3547:

is compact. In particular, this is equivalent to proper discontinuity

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The composition of two morphisms is again a morphism. If a morphism

7365:. This is useful, for instance, in studying the action of the large 4208:. The coinvariant terminology and notation are used particularly in 8405:. Cambridge, UK New York: Cambridge University Press. p. 170. 7951: 5725: 5396: 4063:

if and only if it has exactly one orbit, that is, if there exists

4038:

are equivalent if and only if their orbits are the same, that is,

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With the above understanding, it is very common to avoid writing

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is bijective, then its inverse is also a morphism. In this case

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between the group action functors. In analogy, an action of a

6902:

of the plane act on the set of 2D images and patterns, such as

5862:, which permutes 2, 4, 5 and 3, 6, 8, and fixes 1 and 7. Thus, 8612:, Modern Birkhäuser Classics, Birkhäuser, pp. xxvii+467, 8315:

Lie Groups: An Approach through Invariants and Representations

8014:

as a category with a single object in which every morphism is

5971:

consists only of the identity automorphism, as any element of

4845:

is trivial, the action is said to be faithful (or effective).

4175:, while in algebraic situations it may be called the space of 878:

also on triangles by transforming triangles into triangles.

6093:{\displaystyle |X/G|={\frac {1}{|G|}}\sum _{g\in G}|X^{g}|,} 5994:

A result closely related to the orbit-stabilizer theorem is

4167:

of the action. In geometric situations it may be called the

2601:) if it is both transitive and free. This means that given 5513:

is finite then the orbit-stabilizer theorem, together with

5192:. The condition for two elements to have the same image is 3779:, the symmetry group is the (rotational) icosahedral group 3179:

is wandering. In a dynamical context this is also called a

1787:. Therefore, one may equivalently define a group action of 1576:. This axiom can be shortened even further, and written as 837:

by 0°, 120° and 240° acts on the set of the three vertices.

8018:. A (left) group action is then nothing but a (covariant) 7459:

leaves all points where they were, as does the quaternion

7369:

on a 24-set and in studying symmetry in certain models of

7020:

but has only a trivial action on elements of the subfield

5056:(that is, the set of all conjugates of the subgroup). Let 4194:, by contrast with the invariants (fixed points), denoted 5131:

Orbits and stabilizers are closely related. For a fixed

2047:

when the action being considered is clear from context)

7455:. This is not a faithful action because the quaternion 5046:

to each other. Thus, to each orbit, we can associate a

4795:

if and only if all stabilizers are trivial. The kernel

4216:, which use the same superscript/subscript convention. 8682:, Princeton lecture notes, p. 175, archived from 3267:

is wandering and free but not properly discontinuous.

2857:

when the action on tuples without repeated entries in

2827:

of tuples without repeated entries is transitive. For

2425:. This is a much stronger property than faithfulness. 6004: 5523: 5198: 4707: 3892: 3111:

is called a point of discontinuity for the action of

2103: 2063: 2002:{\displaystyle \alpha (\alpha (x,g),h)=\alpha (x,gh)} 1941: 1894: 1840: 1751:

From these two axioms, it follows that for any fixed

1689: 1654: 1449: 1401: 1392:. The identity and compatibility relations then read 1271:{\displaystyle \alpha (g,\alpha (h,x))=\alpha (gh,x)} 1210: 1163: 1105: 490: 465: 428: 8318:. Springer Science & Business Media. p. 5. 7932:

With this notion of morphism, the collection of all

6538:, given by the action of 1. Similarly, an action of 5471:

of cosets for the stabilizer subgroup and the orbit

5010:. An opposite inclusion follows similarly by taking 4779:, though typically not a normal one. The action of 3715:, i.e. action which are smooth on the whole space. 2155:{\displaystyle (x{\cdot }g){\cdot }h=x{\cdot }(gh)} 1781:, with inverse bijection the corresponding map for 1741:{\displaystyle g{\cdot }(h{\cdot }x)=(gh){\cdot }x} 6825: 6092: 5616: 5354: 4761: 3941: 3333:. Actions with this property are sometimes called 2154: 2083: 2001: 1921: 1867: 1740: 1674: 1553: 1429: 1270: 1190: 1132: 498: 473: 436: 7873:by left multiplication on the first coordinate. ( 7605:The notion of group action can be encoded by the 7050:, that is, intermediate field extensions between 3942:{\displaystyle G{\cdot }x=\{g{\cdot }x:g\in G\}.} 2473:cannot act faithfully on a set of size less than 8699:Three-dimensional geometry and topology. Vol. 1. 8095:acting on objects of their respective category. 7433:is a counterclockwise rotation through an angle 2621:in the definition of transitivity is unique. If 8032:, and a group representation is a functor from 7979:of some category, and then define an action on 7898:action is isomorphic to left multiplication by 2460:. This is not always the case, for example the 34:Transformations induced by a mathematical group 4801:of the homomorphism with the symmetric group, 4762:{\displaystyle G_{x}=\{g\in G:g{\cdot }x=x\}.} 6693:The automorphism group of a vector space (or 5983:| = 8 ⋅ 3 ⋅ 2 ⋅ 1 = 48 5801:. Applying the theorem now to the stabilizer 2821:. In other words the action on the subset of 2627:is acted upon simply transitively by a group 1868:{\displaystyle \alpha \colon X\times G\to X,} 1133:{\displaystyle \alpha \colon G\times X\to X,} 800: 8: 8679:The geometry and topology of three-manifolds 7985:as a monoid homomorphism into the monoid of 6528:uniquely determines and is determined by an 4753: 4721: 3933: 3907: 2413:. In other words, no non-trivial element of 1349:, so that, instead, one has a collection of 7904:on the set of left cosets of some subgroup 6224:; that is, every group element induces the 4565:, and the higher cohomology groups are the 6382:to a subgroup of the permutation group of 6133:, the set of formal differences of finite 5877:. Applying the theorem a third time gives 5624:in other words the length of the orbit of 5329: 5325: 5295: 5291: 5260: 5256: 5230: 5226: 3278:of a locally simply connected space on an 2240:can be considered as a left action of its 807: 793: 245: 71: 36: 8299: 7801:; for all practical purposes, isomorphic 7130:is defined to be the state of the system 6082: 6076: 6067: 6055: 6043: 6035: 6029: 6021: 6013: 6005: 6003: 5707:-invariant elements are congruent modulo 5630:times the order of its stabilizer is the 5606: 5600: 5591: 5586: 5581: 5573: 5561: 5556: 5552: 5538: 5524: 5522: 5343: 5319: 5300: 5277: 5265: 5248: 5234: 5197: 5115:. A maximal orbit type is often called a 4739: 4712: 4706: 3913: 3896: 3891: 2135: 2121: 2110: 2102: 2067: 2062: 1940: 1893: 1839: 1730: 1704: 1693: 1688: 1658: 1653: 1533: 1508: 1495: 1467: 1454: 1448: 1406: 1400: 1209: 1162: 1104: 492: 491: 489: 467: 466: 464: 430: 429: 427: 8610:Hyperbolic manifolds and discrete groups 8263: 8251: 8239: 5695:-invariant elements. More specifically, 5617:{\displaystyle |G\cdot x|==|G|/|G_{x}|,} 3810:corresponds to the tetrahedron to which 3753:-invariant submodules. It is said to be 2842:is well-studied in finite group theory. 2727:with pairwise distinct entries (that is 8593:. Textbooks in mathematics. CRC Press. 8568:An Introduction to the Theory of Groups 8449: 8437: 8425: 8275: 8172: 8102: 7158:The additive group of the real numbers 3127:such that there are only finitely many 3073:such that there are only finitely many 363: 129: 39: 8287: 8216:"Definition:Right Group Action Axioms" 7820:action is isomorphic to the action of 6312:, left multiplication is an action of 6243:, left multiplication is an action of 5665:elements. Since each orbit has either 4974:to both sides of this equality yields 4427:. Conversely, any invariant subset of 4411:Every orbit is an invariant subset of 3711:. There is a well-developed theory of 365:Classification of finite simple groups 8515:Eie, Minking; Chang, Shou-Te (2010). 8496:Dummit, David; Richard Foote (2003). 7879:can be taken to be the set of orbits 7439:about an axis given by a unit vector 7388:), as a multiplicative group, act on 7298:, we can define an induced action of 7120:describes a state of the system, and 4583:Fixed points and stabilizer subgroups 3511:. This means that given compact sets 2871:The action of the symmetric group of 1878:that satisfies the analogous axioms: 1339:It can be notationally convenient to 7: 7558:" indicates right multiplication by 6716:(including the special linear group 6712:and its subgroups, particularly its 6112:. This result is mainly of use when 4433:is a union of orbits. The action of 3735:acts by linear transformations on a 3683:for the action is the set of points 7629:Morphisms and isomorphisms between 5730:Cubical graph with vertices labeled 4396:. Every subset that is fixed under 2453:) acts faithfully on a set of size 4089:. This is the case if and only if 4026:under this relation; two elements 2957:is at least 2). The action of the 25: 8757:"Action of a group on a manifold" 8574:(4th ed.). Springer-Verlag. 6576:and its subgroups act on the set 6376:this induces an isomorphism from 4886:. Then the two stabilizer groups 3794:of order 12, and the orbit space 3365:if there exists a compact subset 1143:that satisfies the following two 8591:Introduction to abstract algebra 8570:. Graduate Texts in Mathematics 8201:Introduction to abstract algebra 8117: 8105: 7962:We can also consider actions of 7453:quaternions and spatial rotation 6811:of coordinates of the vector in 6554:is equivalent to the data of an 6149:, where addition corresponds to 4689:) is the set of all elements in 3864:can be moved by the elements of 3564:if there exists a neighbourhood 1430:{\displaystyle \alpha _{e}(x)=x} 57: 8813:Representation theory of groups 8091:. All of these are examples of 7946:(in fact, assuming a classical 6925:-sets in which the objects are 3747:if there are no proper nonzero 2351:corresponding to the action is 2218:second. Because of the formula 8543:, Cambridge University Press, 8473:. Cambridge University Press. 8196:This is done, for example, by 6941:-set homomorphisms: functions 6463:, conjugation is an action of 6139:-sets forms a ring called the 6106:is the set of points fixed by 6083: 6068: 6044: 6036: 6022: 6006: 5607: 5592: 5582: 5574: 5567: 5546: 5539: 5525: 5500:. This result is known as the 5326: 5292: 5257: 5227: 5223: 5217: 5208: 5202: 5064:denote the conjugacy class of 2682:elements, and for any pair of 2345:to the group of bijections of 2149: 2140: 2118: 2104: 1996: 1981: 1972: 1963: 1951: 1945: 1922:{\displaystyle \alpha (x,e)=x} 1910: 1898: 1856: 1727: 1718: 1712: 1698: 1548: 1542: 1523: 1517: 1514: 1488: 1482: 1479: 1473: 1460: 1418: 1412: 1265: 1250: 1241: 1238: 1226: 1214: 1191:{\displaystyle \alpha (e,x)=x} 1179: 1167: 1121: 726:Infinite dimensional Lie group 1: 8697:Thurston, William P. (1997), 8589:Smith, Jonathan D.H. (2008). 8349:, Proposition 6.8.4 on p. 179 7832:given by left multiplication. 7807:-sets are indistinguishable. 6909:The sets acted on by a group 6774:that act on the vector space 5675:elements, there are at least 4872:be a group element such that 3816:sends the chosen tetrahedron. 3406:Actions of topological groups 3206:there are only finitely many 3005:preserved by all elements of 2264:Notable properties of actions 2210:second. For a right action, 1795:as a group homomorphism from 8517:A Course on Abstract Algebra 8387:A Course on Abstract Algebra 8183:A Course on Abstract Algebra 7958:Variants and generalizations 7601:Groupoid § Group action 7114:is in the phase space, then 5762:acts on the set of vertices 5399:for the stabilizer subgroup 2084:{\displaystyle x{\cdot }e=x} 1675:{\displaystyle e{\cdot }x=x} 865:, and to say that one has a 499:{\displaystyle \mathbb {Z} } 474:{\displaystyle \mathbb {Z} } 437:{\displaystyle \mathbb {Z} } 8808:Group actions (mathematics) 8762:Encyclopedia of Mathematics 7810:Some example isomorphisms: 7595:Group actions and groupoids 7580:; more generally, it is an 7038:correspond to subfields of 6743:, special orthogonal group 4204:while the invariants are a 3777:compound of five tetrahedra 3743:, the action is said to be 3117:if there is an open subset 2975:primitive permutation group 2635:principal homogeneous space 2274:be a group acting on a set 1320:together with an action of 224:List of group theory topics 8834: 8676:Thurston, William (1980), 8660:10.1142/9789811286018_0005 8608:Kapovich, Michael (2009), 7598: 7451:is the same rotation; see 7394:: for any such quaternion 7065:The additive group of the 6895:is of particular interest. 5647:be a group of prime order 5171:. By definition the image 4022:. The orbits are then the 3852:is the set of elements in 3722: 3409: 2972: 2840:multiply transitive groups 1371:, with one transformation 935:. Similarly, the group of 26: 8725:10.1515/9783110858372.312 8713:tom Dieck, Tammo (1987), 8312:Procesi, Claudio (2007). 8156:Young–Deruyts development 8040:category of vector spaces 6789:The general linear group 6700:The general linear group 6686:is a group action called 6584:by permuting its elements 5182:of this map is the orbit 4200:: the coinvariants are a 4149:(or, less frequently, as 4127:The set of all orbits of 3628:The action is said to be 3457:The action is said to be 2889:up to the cardinality of 1799:into the symmetric group 8652:Starting Category Theory 8634:Maskit, Bernard (1988), 8384:Eie & Chang (2010). 8180:Eie & Chang (2010). 7841:action is isomorphic to 7286:Given a group action of 6153:, and multiplication to 5502:orbit-stabilizer theorem 5125:Orbit-stabilizer theorem 4931:. Proof: by definition, 4402:is also invariant under 3763:of irreducible actions. 3646:is continuous for every 3169:such that the action of 3101:More generally, a point 2863:is sharply transitive. 2487:, the icosahedral group 2374:) if the statement that 2214:acts first, followed by 2206:acts first, followed by 342:Elementary abelian group 219:Glossary of group theory 29:group action (sociology) 8650:Perrone, Paolo (2024), 8566:Rotman, Joseph (1995). 8500:(3rd ed.). Wiley. 8359:Carter, Nathan (2009). 8242:, Definition 3.5.1(iv). 8146:Measurable group action 7954:will even be Boolean). 7182:equal to, for example, 6845:projective linear group 6809:greatest common divisor 6366:contains no nontrivial 3679:, then the subspace of 3677:differentiable manifold 3412:Continuous group action 3154:domain of discontinuity 2877:is transitive, in fact 2516:Transitivity properties 2280:. The action is called 1807:of all bijections from 1382:for each group element 1316:(from the left). A set 1312:is then said to act on 8050:natural transformation 7742:-sets are also called 6094: 5731: 5618: 5356: 4763: 4501:. The set of all such 4408:, but not conversely. 3943: 3817: 3767:Orbits and stabilizers 3759:if it decomposes as a 3450:is continuous for the 3190:properly discontinuous 3023:Topological properties 2540:if for any two points 2432:that any group can be 2156: 2085: 2003: 1923: 1869: 1742: 1676: 1555: 1431: 1272: 1192: 1134: 838: 758:Linear algebraic group 500: 475: 438: 8716:Transformation groups 8403:Geometry and topology 8071:, regular actions of 8042:. A morphism between 8003:group representations 7942:; this category is a 7088:(and in more general 6688:scalar multiplication 6318:on the set of cosets 6095: 5989: 5953:. One also sees that 5746:as pictured, and let 5729: 5619: 5357: 4764: 4559:with coefficients in 4307:(which is equivalent 4159:), and is called the 3983:is defined by saying 3944: 3774: 3352:locally compact space 3344:An action of a group 3015:, the partition into 2502:and the cyclic group 2398:already implies that 2358:The action is called 2202:. For a left action, 2157: 2086: 2004: 1924: 1870: 1763:to itself which maps 1743: 1677: 1556: 1432: 1273: 1193: 1135: 910:to some group (under 822: 501: 476: 439: 8654:, World Scientific, 8519:. World Scientific. 8401:Reid, Miles (2005). 8141:Group with operators 8008:We can view a group 6828:on the points of an 6612:with group of units 6567:The symmetric group 6226:identity permutation 6174:action of any group 6002: 5723:is finite as well). 5521: 5196: 5117:principal orbit type 4705: 4631:is a fixed point of 4133:under the action of 4059:The group action is 3981:equivalence relation 3963:under the action of 3890: 3725:Group representation 3335:freely discontinuous 3292:has a neighbourhood 3272:deck transformations 3162:-stable open subset 2951:if the dimension of 2949:special linear group 2932:general linear group 2919:-transitive but not 2907:, the action of the 2883:-transitive for any 2633:then it is called a 2335:. Equivalently, the 2101: 2061: 1939: 1892: 1838: 1759:, the function from 1687: 1652: 1629:can be shortened to 1570:function composition 1447: 1399: 1208: 1161: 1103: 972:general linear group 958:A group action on a 925:Euclidean isometries 914:) of functions from 912:function composition 855:function composition 488: 463: 426: 8471:Finite Group Theory 8467:Aschbacher, Michael 8361:Visual Group Theory 8077:algebraic varieties 7584:in the category of 7496:whose elements are 7086:classical mechanics 6360:. In particular if 5143:, consider the map 4860:be two elements in 4816:of the stabilizers 4665:stabilizer subgroup 4625:, it is said that " 4024:equivalence classes 3632:if the orbital map 3630:strongly continuous 2836:2-transitive groups 2031:often shortened to 1574:commutative diagram 987:invertible matrices 985:, the group of the 955:of the polyhedron. 857:; for example, the 132:Group homomorphisms 42:Algebraic structure 8776:Weisstein, Eric W. 8540:Algebraic Topology 8061:continuous actions 7944:Grothendieck topos 7582:exponential object 7502:-equivariant maps 7484:, there is a left 7014:acts on the field 6904:wallpaper patterns 6090: 6066: 5732: 5701:and the number of 5632:order of the group 5614: 5515:Lagrange's theorem 5352: 5084:if the stabilizer 4812:, is given by the 4759: 3939: 3818: 3693:such that the map 3037:and the action of 2934:of a vector space 2930:The action of the 2591:sharply transitive 2152: 2081: 1999: 1919: 1865: 1821:right group action 1815:Right group action 1738: 1672: 1551: 1427: 1268: 1188: 1130: 1074:is a set, then a ( 902:group homomorphism 839: 833:consisting of the 608:Special orthogonal 496: 471: 434: 315:Lagrange's theorem 8734:978-3-11-009745-0 8669:978-981-12-8600-1 8619:978-0-8176-4912-8 8600:978-1-4200-6371-4 8550:978-0-521-79540-1 8526:978-981-4271-88-2 8480:978-0-521-78675-1 8290:, II.A.1, II.A.2. 8067:of Lie groups on 8005:in this fashion. 7892:Every transitive 7526:-action given by 7371:finite geometries 7341:for every subset 7136:seconds later if 7090:dynamical systems 6469:on conjugates of 6155:Cartesian product 6051: 6049: 5928:. Any element of 5848:. Any element of 5790:| = 8 | 5685:orbits of length 5070:. Then the orbit 5050:of a subgroup of 4681:(also called the 4327:also operates on 4321:). In that case, 4220:Invariant subsets 3979:. The associated 3820:Consider a group 3787:tetrahedral group 3713:Lie group actions 3560:It is said to be 3424:topological group 3276:fundamental group 3035:topological space 2969:Primitive actions 2909:alternating group 2777:) there exists a 2585:simply transitive 2419:fixes a point of 2165: 2164: 2012: 2011: 1281: 1280: 1051:Left group action 817: 816: 392: 391: 274:Alternating group 231: 230: 16:(Redirected from 8825: 8789: 8788: 8770: 8745: 8709: 8693: 8692: 8691: 8672: 8646: 8630: 8604: 8585: 8561: 8530: 8511: 8498:Abstract Algebra 8492: 8453: 8452:, pp. 69–71 8447: 8441: 8440:, pp. 36–39 8435: 8429: 8423: 8417: 8416: 8398: 8392: 8391: 8381: 8375: 8374: 8356: 8350: 8343: 8337: 8336: 8334: 8332: 8309: 8303: 8297: 8291: 8285: 8279: 8273: 8267: 8261: 8255: 8249: 8243: 8237: 8231: 8230: 8228: 8226: 8212: 8206: 8205: 8194: 8188: 8187: 8177: 8121: 8109: 8073:algebraic groups 8069:smooth manifolds 8048:-sets is then a 8047: 8037: 8030:category of sets 8027: 8013: 8000: 7994: 7984: 7978: 7968:semigroup action 7937: 7927: 7921: 7915: 7909: 7903: 7897: 7888: 7878: 7872: 7862: 7857:is some set and 7856: 7850: 7840: 7831: 7825: 7819: 7806: 7796: 7790: 7784: 7772: 7766: 7753: 7745:equivariant maps 7741: 7735: 7729: 7723: 7717: 7711: 7684: 7670: 7664: 7654: 7648: 7642: 7624: 7589: 7579: 7569: 7563: 7557: 7550: 7525: 7520:, and with left 7519: 7501: 7495: 7489: 7483: 7477: 7471: 7462: 7458: 7450: 7444: 7438: 7432: 7412: 7393: 7364: 7358: 7352: 7346: 7340: 7313: 7303: 7297: 7291: 7282: 7267: 7253: 7235: 7221: 7210: 7196: 7181: 7165: 7154: 7148: 7141: 7135: 7129: 7119: 7113: 7107: 7101: 7094:time translation 7075: 7061: 7055: 7049: 7043: 7037: 7025: 7019: 7013: 6993: 6987: 6981: 6954: 6940: 6930: 6924: 6914: 6894: 6879: 6871: 6860:projective space 6857: 6816: 6806: 6800: 6785: 6779: 6769: 6757:symplectic group 6754: 6742: 6730:orthogonal group 6727: 6711: 6685: 6632: 6618: 6611: 6605: 6599:coordinate space 6583: 6575: 6563: 6553: 6547: 6537: 6527: 6521: 6512: 6506: 6500: 6494: 6488: 6474: 6468: 6462: 6456: 6447: 6438:; it satisfies ( 6437: 6424: 6410: 6404: 6399:is an action of 6394: 6381: 6375: 6368:normal subgroups 6365: 6359: 6353: 6347: 6341: 6327: 6317: 6311: 6305: 6296: 6289:Cayley's theorem 6286: 6280: 6274: 6268: 6254: 6248: 6242: 6233: 6223: 6217: 6211: 6205: 6199: 6185: 6179: 6172: 6171: 6148: 6138: 6132: 6123: 6117: 6111: 6105: 6099: 6097: 6096: 6091: 6086: 6081: 6080: 6071: 6065: 6050: 6048: 6047: 6039: 6030: 6025: 6017: 6009: 5996:Burnside's lemma 5990:Burnside's lemma 5984: 5982: 5976: 5970: 5952: 5950: 5933: 5927: 5925: 5907: 5892: 5876: 5874: 5861: 5853: 5847: 5845: 5831: 5820: 5810:, we can obtain 5809: 5800: 5798: 5789: 5780: 5773: 5765: 5761: 5751: 5722: 5712: 5706: 5700: 5694: 5688: 5684: 5674: 5668: 5664: 5658: 5653:acting on a set 5652: 5646: 5629: 5623: 5621: 5620: 5615: 5610: 5605: 5604: 5595: 5590: 5585: 5577: 5566: 5565: 5542: 5528: 5512: 5499: 5480: 5470: 5456:between the set 5451: 5445: 5435: 5429: 5423: 5409: 5395:lie in the same 5394: 5388: 5380: 5362:In other words, 5361: 5359: 5358: 5353: 5348: 5347: 5324: 5323: 5308: 5307: 5281: 5273: 5272: 5252: 5238: 5191: 5181: 5170: 5156: 5142: 5136: 5127: 5126: 5114: 5106: 5100: 5094: 5083: 5075: 5069: 5063: 5055: 5038: 5024: 5009: 4991: 4973: 4967: 4945: 4930: 4907: 4896: 4885: 4871: 4865: 4859: 4853: 4844: 4838: 4832: 4826: 4811: 4800: 4790: 4784: 4778: 4768: 4766: 4765: 4760: 4743: 4717: 4716: 4700: 4694: 4680: 4675:with respect to 4674: 4667: 4666: 4660: 4654: 4648: 4642: 4636: 4630: 4624: 4610: 4604: 4598: 4592: 4578: 4567:derived functors 4564: 4558: 4548: 4540: 4533: 4527: 4518: 4512: 4506: 4500: 4490: 4476: 4466: 4457: 4444: 4438: 4432: 4422: 4416: 4407: 4401: 4395: 4389: 4383: 4377: 4371: 4357: 4348: 4342: 4332: 4326: 4320: 4306: 4292: 4286:invariant under 4283: 4277: 4250:denotes the set 4249: 4239: 4229: 4210:group cohomology 4199: 4193: 4181: 4180: 4173: 4172: 4165: 4164: 4158: 4148: 4138: 4132: 4123: 4117: 4111: 4102: 4088: 4074: 4068: 4055: 4037: 4031: 4021: 4007: 4001: 3992: 3978: 3968: 3962: 3956: 3948: 3946: 3945: 3940: 3917: 3900: 3885: 3875: 3869: 3863: 3857: 3851: 3845: 3838: 3837: 3831: 3826:acting on a set 3825: 3815: 3809: 3803: 3793: 3784: 3752: 3741:commutative ring 3734: 3706: 3692: 3674: 3664: 3655: 3645: 3624: 3604: 3594: 3580: 3569: 3552: 3546: 3531: 3521: 3506: 3482: 3463: 3462: 3452:product topology 3449: 3431: 3421: 3401: 3388: 3374: 3358: 3349: 3339:free regular set 3332: 3312: 3297: 3291: 3266: 3243: 3242:∖ {(0, 0)} 3236: 3230: 3215: 3205: 3178: 3174: 3168: 3161: 3151: 3136: 3126: 3116: 3110: 3097: 3082: 3072: 3063: 3042: 3032: 3010: 3004: 2990: 2984: 2959:orthogonal group 2956: 2946: 2939: 2926: 2918: 2906: 2901:has cardinality 2900: 2894: 2888: 2882: 2876: 2862: 2856: 2855: 2852: 2833: 2826: 2820: 2810: 2786: 2776: 2766: 2746: 2726: 2687: 2681: 2675: 2669: 2668: 2665: 2659:, the action is 2658: 2648: 2642: 2632: 2626: 2620: 2614: 2599: 2598: 2587: 2586: 2577: 2563: 2553: 2538: 2537: 2531: 2525: 2511: 2501: 2486: 2476: 2472: 2459: 2452: 2449:(of cardinality 2448: 2430:Cayley's theorem 2424: 2418: 2412: 2397: 2387: 2372:fixed-point free 2364: 2363: 2350: 2344: 2334: 2319: 2309: 2294: 2293: 2286: 2285: 2279: 2273: 2252: 2248: 2239: 2235: 2231: 2217: 2213: 2209: 2205: 2201: 2197: 2188: 2184: 2180: 2176: 2172: 2161: 2159: 2158: 2153: 2139: 2125: 2114: 2090: 2088: 2087: 2082: 2071: 2052: 2051: 2046: 2036: 2030: 2008: 2006: 2005: 2000: 1928: 1926: 1925: 1920: 1883: 1882: 1874: 1872: 1871: 1866: 1830: 1826: 1810: 1806: 1798: 1794: 1790: 1786: 1776: 1766: 1762: 1758: 1754: 1747: 1745: 1744: 1739: 1734: 1708: 1697: 1681: 1679: 1678: 1673: 1662: 1644: 1638: 1628: 1613: 1604: 1567: 1560: 1558: 1557: 1552: 1541: 1540: 1513: 1512: 1500: 1499: 1472: 1471: 1459: 1458: 1436: 1434: 1433: 1428: 1411: 1410: 1391: 1381: 1370: 1348: 1331: 1323: 1319: 1315: 1311: 1304: 1300: 1296: 1292: 1288: 1277: 1275: 1274: 1269: 1197: 1195: 1194: 1189: 1152: 1151: 1139: 1137: 1136: 1131: 1092: 1088: 1084: 1073: 1069: 1065:identity element 1058: 1033: 1023: 1006: 997: 984: 919: 909: 899: 890: 832: 809: 802: 795: 751:Algebraic groups 524:Hyperbolic group 514:Arithmetic group 505: 503: 502: 497: 495: 480: 478: 477: 472: 470: 443: 441: 440: 435: 433: 356:Schur multiplier 310:Cauchy's theorem 298:Quaternion group 246: 72: 61: 48: 37: 21: 18:Transitive group 8833: 8832: 8828: 8827: 8826: 8824: 8823: 8822: 8793: 8792: 8774: 8773: 8755: 8752: 8735: 8712: 8696: 8689: 8687: 8675: 8670: 8649: 8636:Kleinian groups 8633: 8620: 8607: 8601: 8588: 8582: 8565: 8551: 8533: 8527: 8514: 8508: 8495: 8481: 8465: 8462: 8457: 8456: 8448: 8444: 8436: 8432: 8424: 8420: 8413: 8400: 8399: 8395: 8383: 8382: 8378: 8371: 8358: 8357: 8353: 8344: 8340: 8330: 8328: 8326: 8311: 8310: 8306: 8298: 8294: 8286: 8282: 8274: 8270: 8262: 8258: 8250: 8246: 8238: 8234: 8224: 8222: 8214: 8213: 8209: 8197: 8195: 8191: 8179: 8178: 8174: 8169: 8164: 8132: 8125: 8122: 8113: 8110: 8101: 8059:In addition to 8043: 8033: 8023: 8009: 7996: 7990: 7980: 7974: 7960: 7933: 7923: 7917: 7911: 7905: 7899: 7893: 7880: 7874: 7864: 7858: 7852: 7842: 7836: 7827: 7821: 7815: 7802: 7792: 7786: 7780: 7768: 7762: 7749: 7737: 7736:. Morphisms of 7731: 7725: 7719: 7713: 7686: 7672: 7666: 7660: 7650: 7644: 7638: 7635: 7612: 7603: 7597: 7585: 7571: 7565: 7559: 7552: 7544: 7527: 7521: 7503: 7497: 7491: 7485: 7479: 7473: 7467: 7460: 7456: 7446: 7440: 7434: 7414: 7395: 7389: 7360: 7354: 7348: 7342: 7315: 7309: 7299: 7293: 7287: 7269: 7255: 7237: 7223: 7212: 7198: 7183: 7167: 7159: 7150: 7149:seconds ago if 7143: 7142:is positive or 7137: 7131: 7121: 7115: 7109: 7103: 7097: 7069: 7057: 7051: 7045: 7039: 7027: 7026:. Subgroups of 7021: 7015: 7005: 7003:field extension 6989: 6983: 6956: 6942: 6936: 6926: 6920: 6910: 6888: 6873: 6862: 6847: 6812: 6802: 6790: 6781: 6775: 6759: 6744: 6732: 6717: 6701: 6683: 6674: 6667: 6660: 6651: 6644: 6634: 6620: 6613: 6607: 6601: 6577: 6574: 6568: 6559: 6549: 6539: 6533: 6523: 6517: 6508: 6502: 6496: 6490: 6476: 6470: 6464: 6458: 6452: 6451:In every group 6439: 6426: 6412: 6406: 6400: 6390: 6389:In every group 6377: 6371: 6361: 6355: 6349: 6343: 6329: 6319: 6313: 6307: 6301: 6300:In every group 6292: 6282: 6276: 6270: 6256: 6250: 6244: 6238: 6237:In every group 6229: 6219: 6213: 6207: 6201: 6187: 6181: 6175: 6169: 6168: 6163: 6144: 6134: 6128: 6127:Fixing a group 6119: 6113: 6107: 6101: 6072: 6034: 6000: 5999: 5992: 5980: 5978: 5972: 5969: 5965: 5961: 5954: 5948: 5944: 5937: 5935: 5929: 5924: 5920: 5916: 5909: 5905: 5901: 5894: 5893:| = | 5891: 5887: 5880: 5878: 5872: 5865: 5863: 5855: 5849: 5844: 5840: 5833: 5829: 5822: 5821:| = | 5819: 5813: 5811: 5808: 5802: 5797: 5791: 5788: 5782: 5775: 5774:| = | 5769: 5767: 5763: 5757: 5747: 5742:. Consider the 5718: 5708: 5702: 5696: 5690: 5686: 5676: 5670: 5666: 5660: 5654: 5648: 5642: 5625: 5596: 5557: 5519: 5518: 5508: 5490: 5482: 5472: 5469: 5457: 5447: 5437: 5431: 5425: 5414: 5408: 5400: 5390: 5384: 5363: 5339: 5315: 5296: 5261: 5194: 5193: 5183: 5172: 5158: 5144: 5138: 5132: 5129: 5124: 5123: 5108: 5102: 5096: 5093: 5085: 5077: 5071: 5065: 5057: 5051: 5048:conjugacy class 5026: 5023: 5011: 5008: 4993: 4975: 4969: 4947: 4946:if and only if 4944: 4932: 4926: 4917: 4909: 4908:are related by 4906: 4898: 4895: 4887: 4873: 4867: 4861: 4855: 4849: 4840: 4834: 4828: 4825: 4817: 4802: 4796: 4786: 4780: 4774: 4708: 4703: 4702: 4696: 4690: 4676: 4670: 4664: 4663: 4656: 4650: 4644: 4638: 4632: 4626: 4612: 4606: 4600: 4594: 4588: 4585: 4574: 4560: 4554: 4544: 4536: 4529: 4523: 4514: 4513:and called the 4508: 4502: 4492: 4478: 4468: 4462: 4453: 4440: 4434: 4428: 4418: 4412: 4403: 4397: 4391: 4385: 4379: 4373: 4359: 4353: 4344: 4338: 4328: 4322: 4308: 4294: 4288: 4279: 4251: 4241: 4235: 4225: 4222: 4195: 4192: 4184: 4178: 4177: 4170: 4169: 4162: 4161: 4150: 4140: 4134: 4128: 4124:is non-empty). 4119: 4113: 4107: 4090: 4076: 4070: 4064: 4039: 4033: 4027: 4009: 4003: 3997: 3996:there exists a 3984: 3974: 3964: 3958: 3952: 3888: 3887: 3877: 3871: 3870:. The orbit of 3865: 3859: 3853: 3847: 3841: 3835: 3834: 3827: 3821: 3811: 3805: 3795: 3789: 3780: 3769: 3748: 3730: 3727: 3721: 3694: 3684: 3670: 3660: 3647: 3633: 3622: 3606: 3596: 3582: 3579: 3571: 3565: 3548: 3533: 3523: 3512: 3484: 3466: 3460: 3459: 3437: 3427: 3417: 3414: 3408: 3393: 3376: 3366: 3354: 3345: 3330: 3314: 3299: 3293: 3283: 3245: 3238: 3232: 3217: 3207: 3197: 3176: 3170: 3163: 3157: 3138: 3128: 3118: 3112: 3102: 3084: 3074: 3068: 3055: 3038: 3028: 3025: 3006: 3000: 2995:if there is no 2986: 2980: 2977: 2971: 2952: 2941: 2935: 2920: 2912: 2902: 2896: 2890: 2884: 2878: 2872: 2869: 2858: 2850: 2847: 2846: 2828: 2822: 2812: 2809: 2800: 2788: 2778: 2768: 2765: 2756: 2748: 2745: 2736: 2728: 2721: 2712: 2705: 2696: 2689: 2683: 2677: 2671: 2663: 2661: 2660: 2653: 2652:For an integer 2644: 2638: 2628: 2622: 2616: 2602: 2596: 2595: 2584: 2583: 2565: 2555: 2554:there exists a 2541: 2535: 2534: 2527: 2521: 2518: 2503: 2492: 2488: 2485: 2481: 2474: 2464: 2454: 2450: 2438: 2420: 2414: 2411: 2399: 2389: 2375: 2361: 2360: 2346: 2340: 2333: 2321: 2311: 2297: 2291: 2290: 2283: 2282: 2275: 2269: 2266: 2250: 2244: 2237: 2233: 2219: 2215: 2211: 2207: 2203: 2199: 2193: 2186: 2182: 2178: 2174: 2170: 2099: 2098: 2095:Compatibility: 2059: 2058: 2038: 2032: 2017: 1937: 1936: 1933:Compatibility: 1890: 1889: 1836: 1835: 1828: 1824: 1817: 1808: 1800: 1796: 1792: 1788: 1782: 1768: 1764: 1760: 1756: 1752: 1685: 1684: 1650: 1649: 1640: 1630: 1615: 1609: 1603: 1594: 1585: 1577: 1565: 1529: 1504: 1491: 1463: 1450: 1445: 1444: 1402: 1397: 1396: 1383: 1380: 1372: 1361: 1353: 1351:transformations 1344: 1329: 1321: 1317: 1313: 1309: 1302: 1298: 1294: 1290: 1286: 1206: 1205: 1202:Compatibility: 1159: 1158: 1101: 1100: 1090: 1086: 1082: 1071: 1067: 1056: 1053: 1048: 1029: 1022: 1014: 1012:symmetric group 1002: 993: 974: 929:Euclidean space 915: 905: 895: 886: 847:transformations 845:, many sets of 831: 827: 813: 784: 783: 772:Abelian variety 765:Reductive group 753: 743: 742: 741: 740: 691: 683: 675: 667: 659: 632:Special unitary 543: 529: 528: 510: 509: 486: 485: 461: 460: 424: 423: 415: 414: 405:Discrete groups 394: 393: 349:Frobenius group 294: 281: 270: 263:Symmetric group 259: 243: 233: 232: 83:Normal subgroup 69: 49: 40: 35: 32: 23: 22: 15: 12: 11: 5: 8831: 8829: 8821: 8820: 8815: 8810: 8805: 8795: 8794: 8791: 8790: 8779:"Group Action" 8771: 8751: 8750:External links 8748: 8747: 8746: 8733: 8710: 8694: 8673: 8668: 8647: 8631: 8618: 8605: 8599: 8586: 8580: 8563: 8549: 8535:Hatcher, Allen 8531: 8525: 8512: 8506: 8493: 8479: 8461: 8458: 8455: 8454: 8450:Perrone (2024) 8442: 8438:Perrone (2024) 8430: 8428:, pp. 7–9 8426:Perrone (2024) 8418: 8411: 8393: 8390:. p. 145. 8376: 8370:978-0883857571 8369: 8351: 8338: 8324: 8304: 8300:tom Dieck 1987 8292: 8280: 8268: 8266:, p. 176. 8256: 8244: 8232: 8207: 8204:. p. 253. 8198:Smith (2008). 8189: 8186:. p. 144. 8171: 8170: 8168: 8165: 8163: 8160: 8159: 8158: 8153: 8148: 8143: 8138: 8131: 8128: 8127: 8126: 8123: 8116: 8114: 8111: 8104: 8100: 8097: 8065:smooth actions 7959: 7956: 7938:-sets forms a 7930: 7929: 7890: 7833: 7814:Every regular 7779:, and the two 7671:is a function 7634: 7627: 7599:Main article: 7596: 7593: 7592: 7591: 7540: 7464: 7413:, the mapping 7374: 7284: 7156: 7063: 6995: 6931:-sets and the 6907: 6896: 6841: 6818: 6787: 6698: 6691: 6679: 6672: 6665: 6656: 6649: 6642: 6619:, the mapping 6595: 6592: 6589:symmetry group 6585: 6570: 6565: 6514: 6507:conjugates of 6457:with subgroup 6449: 6387: 6306:with subgroup 6298: 6235: 6186:is defined by 6162: 6159: 6151:disjoint union 6089: 6085: 6079: 6075: 6070: 6064: 6061: 6058: 6054: 6046: 6042: 6038: 6033: 6028: 6024: 6020: 6016: 6012: 6008: 5991: 5988: 5987: 5986: 5967: 5963: 5959: 5946: 5942: 5922: 5918: 5914: 5903: 5899: 5889: 5885: 5870: 5842: 5838: 5827: 5817: 5806: 5795: 5786: 5764:{1, 2, ..., 8} 5715: 5714: 5613: 5609: 5603: 5599: 5594: 5589: 5584: 5580: 5576: 5572: 5569: 5564: 5560: 5555: 5551: 5548: 5545: 5541: 5537: 5534: 5531: 5527: 5486: 5481:, which sends 5465: 5455: 5404: 5382:if and only if 5351: 5346: 5342: 5338: 5335: 5332: 5328: 5322: 5318: 5314: 5311: 5306: 5303: 5299: 5294: 5290: 5287: 5284: 5280: 5276: 5271: 5268: 5264: 5259: 5255: 5251: 5247: 5244: 5241: 5237: 5233: 5229: 5225: 5222: 5219: 5216: 5213: 5210: 5207: 5204: 5201: 5128: 5121: 5089: 5019: 5004: 4940: 4922: 4913: 4902: 4891: 4821: 4758: 4755: 4752: 4749: 4746: 4742: 4738: 4735: 4732: 4729: 4726: 4723: 4720: 4715: 4711: 4683:isotropy group 4584: 4581: 4549:is the zeroth 4337:the action to 4284:is said to be 4221: 4218: 4214:group homology 4207: 4203: 4188: 4183:, and written 4139:is written as 4106: 3994:if and only if 3938: 3935: 3932: 3929: 3926: 3923: 3920: 3916: 3912: 3909: 3906: 3903: 3899: 3895: 3876:is denoted by 3840:of an element 3768: 3765: 3723:Main article: 3720: 3719:Linear actions 3717: 3618: 3575: 3555:discrete group 3410:Main article: 3407: 3404: 3391:quotient space 3326: 3280:covering space 3270:The action by 3188:The action is 3050:The action is 3045:homeomorphisms 3024: 3021: 2979:The action of 2973:Main article: 2970: 2967: 2927:-transitive. 2868: 2865: 2805: 2796: 2761: 2752: 2741: 2732: 2717: 2710: 2701: 2694: 2581:The action is 2520:The action of 2517: 2514: 2490: 2483: 2407: 2329: 2265: 2262: 2242:opposite group 2167: 2166: 2163: 2162: 2151: 2148: 2145: 2142: 2138: 2134: 2131: 2128: 2124: 2120: 2117: 2113: 2109: 2106: 2096: 2092: 2091: 2080: 2077: 2074: 2070: 2066: 2056: 2014: 2013: 2010: 2009: 1998: 1995: 1992: 1989: 1986: 1983: 1980: 1977: 1974: 1971: 1968: 1965: 1962: 1959: 1956: 1953: 1950: 1947: 1944: 1934: 1930: 1929: 1918: 1915: 1912: 1909: 1906: 1903: 1900: 1897: 1887: 1876: 1875: 1864: 1861: 1858: 1855: 1852: 1849: 1846: 1843: 1831:is a function 1816: 1813: 1749: 1748: 1737: 1733: 1729: 1726: 1723: 1720: 1717: 1714: 1711: 1707: 1703: 1700: 1696: 1692: 1682: 1671: 1668: 1665: 1661: 1657: 1599: 1590: 1581: 1562: 1561: 1550: 1547: 1544: 1539: 1536: 1532: 1528: 1525: 1522: 1519: 1516: 1511: 1507: 1503: 1498: 1494: 1490: 1487: 1484: 1481: 1478: 1475: 1470: 1466: 1462: 1457: 1453: 1438: 1437: 1426: 1423: 1420: 1417: 1414: 1409: 1405: 1376: 1357: 1283: 1282: 1279: 1278: 1267: 1264: 1261: 1258: 1255: 1252: 1249: 1246: 1243: 1240: 1237: 1234: 1231: 1228: 1225: 1222: 1219: 1216: 1213: 1203: 1199: 1198: 1187: 1184: 1181: 1178: 1175: 1172: 1169: 1166: 1156: 1141: 1140: 1129: 1126: 1123: 1120: 1117: 1114: 1111: 1108: 1052: 1049: 1047: 1044: 1018: 964:representation 863:abstract group 829: 815: 814: 812: 811: 804: 797: 789: 786: 785: 782: 781: 779:Elliptic curve 775: 774: 768: 767: 761: 760: 754: 749: 748: 745: 744: 739: 738: 735: 732: 728: 724: 723: 722: 717: 715:Diffeomorphism 711: 710: 705: 700: 694: 693: 689: 685: 681: 677: 673: 669: 665: 661: 657: 652: 651: 640: 639: 628: 627: 616: 615: 604: 603: 592: 591: 580: 579: 572:Special linear 568: 567: 560:General linear 556: 555: 550: 544: 535: 534: 531: 530: 527: 526: 521: 516: 508: 507: 494: 482: 469: 456: 454:Modular groups 452: 451: 450: 445: 432: 416: 413: 412: 407: 401: 400: 399: 396: 395: 390: 389: 388: 387: 382: 377: 374: 368: 367: 361: 360: 359: 358: 352: 351: 345: 344: 339: 330: 329: 327:Hall's theorem 324: 322:Sylow theorems 318: 317: 312: 304: 303: 302: 301: 295: 290: 287:Dihedral group 283: 282: 277: 271: 266: 260: 255: 244: 239: 238: 235: 234: 229: 228: 227: 226: 221: 213: 212: 211: 210: 205: 200: 195: 190: 185: 180: 178:multiplicative 175: 170: 165: 160: 152: 151: 150: 149: 144: 136: 135: 127: 126: 125: 124: 122:Wreath product 119: 114: 109: 107:direct product 101: 99:Quotient group 93: 92: 91: 90: 85: 80: 70: 67: 66: 63: 62: 54: 53: 33: 24: 14: 13: 10: 9: 6: 4: 3: 2: 8830: 8819: 8816: 8814: 8811: 8809: 8806: 8804: 8801: 8800: 8798: 8786: 8785: 8780: 8777: 8772: 8768: 8764: 8763: 8758: 8754: 8753: 8749: 8744: 8740: 8736: 8730: 8726: 8722: 8718: 8717: 8711: 8708: 8704: 8700: 8695: 8686:on 2020-07-27 8685: 8681: 8680: 8674: 8671: 8665: 8661: 8657: 8653: 8648: 8645: 8641: 8637: 8632: 8629: 8625: 8621: 8615: 8611: 8606: 8602: 8596: 8592: 8587: 8583: 8581:0-387-94285-8 8577: 8573: 8569: 8564: 8560: 8556: 8552: 8546: 8542: 8541: 8536: 8532: 8528: 8522: 8518: 8513: 8509: 8507:0-471-43334-9 8503: 8499: 8494: 8490: 8486: 8482: 8476: 8472: 8468: 8464: 8463: 8459: 8451: 8446: 8443: 8439: 8434: 8431: 8427: 8422: 8419: 8414: 8412:9780521613255 8408: 8404: 8397: 8394: 8389: 8388: 8380: 8377: 8372: 8366: 8362: 8355: 8352: 8348: 8342: 8339: 8327: 8325:9780387289298 8321: 8317: 8316: 8308: 8305: 8301: 8296: 8293: 8289: 8284: 8281: 8277: 8272: 8269: 8265: 8264:Thurston 1980 8260: 8257: 8253: 8252:Kapovich 2009 8248: 8245: 8241: 8240:Thurston 1997 8236: 8233: 8221: 8217: 8211: 8208: 8203: 8202: 8193: 8190: 8185: 8184: 8176: 8173: 8166: 8161: 8157: 8154: 8152: 8151:Monoid action 8149: 8147: 8144: 8142: 8139: 8137: 8134: 8133: 8129: 8120: 8115: 8108: 8103: 8098: 8096: 8094: 8093:group objects 8090: 8086: 8085:group schemes 8082: 8078: 8074: 8070: 8066: 8062: 8057: 8055: 8051: 8046: 8041: 8036: 8031: 8026: 8021: 8017: 8012: 8006: 8004: 7999: 7993: 7988: 7987:endomorphisms 7983: 7977: 7971: 7969: 7965: 7957: 7955: 7953: 7949: 7945: 7941: 7936: 7926: 7920: 7914: 7908: 7902: 7896: 7891: 7887: 7883: 7877: 7871: 7867: 7861: 7855: 7849: 7845: 7839: 7834: 7830: 7824: 7818: 7813: 7812: 7811: 7808: 7805: 7800: 7795: 7789: 7783: 7778: 7777: 7773:is called an 7771: 7765: 7759: 7757: 7752: 7747: 7746: 7740: 7734: 7728: 7722: 7716: 7709: 7705: 7701: 7697: 7693: 7689: 7683: 7679: 7675: 7669: 7663: 7658: 7653: 7647: 7641: 7632: 7628: 7626: 7623: 7619: 7615: 7611: 7610: 7602: 7594: 7588: 7583: 7578: 7574: 7568: 7562: 7556: 7548: 7543: 7538: 7534: 7530: 7524: 7518: 7514: 7510: 7506: 7500: 7494: 7488: 7482: 7476: 7470: 7465: 7454: 7449: 7443: 7437: 7431: 7428: 7425: 7421: 7417: 7410: 7406: 7402: 7398: 7392: 7387: 7383: 7379: 7375: 7372: 7368: 7367:Mathieu group 7363: 7357: 7351: 7345: 7338: 7334: 7330: 7326: 7322: 7318: 7314:, by setting 7312: 7307: 7302: 7296: 7290: 7285: 7280: 7276: 7272: 7266: 7262: 7258: 7252: 7248: 7244: 7240: 7234: 7230: 7226: 7219: 7215: 7209: 7205: 7201: 7194: 7190: 7186: 7179: 7175: 7171: 7163: 7157: 7153: 7147: 7140: 7134: 7128: 7124: 7118: 7112: 7106: 7100: 7095: 7091: 7087: 7084:" systems in 7083: 7079: 7073: 7068: 7064: 7060: 7054: 7048: 7044:that contain 7042: 7035: 7031: 7024: 7018: 7012: 7008: 7004: 7000: 6996: 6992: 6986: 6979: 6975: 6971: 6967: 6963: 6959: 6953: 6949: 6945: 6939: 6934: 6929: 6923: 6918: 6915:comprise the 6913: 6908: 6905: 6901: 6897: 6892: 6887: 6883: 6877: 6869: 6865: 6861: 6855: 6851: 6846: 6842: 6839: 6835: 6831: 6827: 6823: 6819: 6815: 6810: 6805: 6798: 6794: 6788: 6784: 6778: 6773: 6767: 6763: 6758: 6752: 6748: 6740: 6736: 6731: 6725: 6721: 6715: 6714:Lie subgroups 6709: 6705: 6699: 6696: 6692: 6689: 6682: 6678: 6671: 6664: 6659: 6655: 6648: 6641: 6637: 6631: 6627: 6623: 6616: 6610: 6606:over a field 6604: 6600: 6596: 6593: 6590: 6586: 6581: 6573: 6566: 6562: 6557: 6552: 6546: 6542: 6536: 6531: 6526: 6520: 6516:An action of 6515: 6511: 6505: 6499: 6493: 6487: 6483: 6479: 6473: 6467: 6461: 6455: 6450: 6446: 6442: 6436: 6433: 6429: 6423: 6419: 6415: 6409: 6403: 6398: 6393: 6388: 6385: 6380: 6374: 6369: 6364: 6358: 6352: 6346: 6340: 6336: 6332: 6326: 6322: 6316: 6310: 6304: 6299: 6295: 6290: 6285: 6279: 6273: 6267: 6263: 6259: 6253: 6247: 6241: 6236: 6232: 6227: 6222: 6216: 6210: 6204: 6198: 6194: 6190: 6184: 6178: 6173: 6165: 6164: 6160: 6158: 6156: 6152: 6147: 6142: 6141:Burnside ring 6137: 6131: 6125: 6122: 6116: 6110: 6104: 6087: 6077: 6073: 6062: 6059: 6056: 6052: 6040: 6031: 6026: 6018: 6014: 6010: 5997: 5975: 5958: 5941: 5932: 5913: 5908:| | 5898: 5884: 5869: 5859: 5852: 5837: 5832:| | 5826: 5816: 5805: 5794: 5785: 5781:| | 5778: 5772: 5760: 5755: 5750: 5745: 5744:cubical graph 5741: 5737: 5734: 5733: 5728: 5724: 5721: 5711: 5705: 5699: 5693: 5683: 5679: 5673: 5663: 5657: 5651: 5645: 5640: 5637: 5636: 5635: 5633: 5628: 5611: 5601: 5597: 5587: 5578: 5570: 5562: 5558: 5553: 5549: 5543: 5535: 5532: 5529: 5516: 5511: 5505: 5503: 5498: 5494: 5489: 5485: 5479: 5475: 5468: 5464: 5460: 5453: 5450: 5444: 5440: 5434: 5428: 5421: 5417: 5413: 5407: 5403: 5398: 5393: 5387: 5383: 5378: 5374: 5370: 5366: 5349: 5344: 5340: 5336: 5333: 5330: 5320: 5316: 5312: 5309: 5304: 5301: 5297: 5288: 5285: 5282: 5278: 5274: 5269: 5266: 5262: 5253: 5249: 5245: 5242: 5239: 5235: 5231: 5220: 5214: 5211: 5205: 5199: 5190: 5186: 5179: 5175: 5169: 5165: 5161: 5155: 5151: 5147: 5141: 5135: 5122: 5120: 5118: 5112: 5105: 5099: 5092: 5088: 5081: 5074: 5068: 5061: 5054: 5049: 5045: 5040: 5037: 5033: 5029: 5022: 5018: 5014: 5007: 5003: 4999: 4996: 4990: 4986: 4982: 4979: 4972: 4966: 4962: 4958: 4954: 4950: 4943: 4939: 4935: 4929: 4925: 4921: 4916: 4912: 4905: 4901: 4894: 4890: 4884: 4880: 4876: 4870: 4864: 4858: 4852: 4846: 4843: 4837: 4831: 4824: 4820: 4815: 4809: 4805: 4799: 4794: 4789: 4783: 4777: 4772: 4756: 4750: 4747: 4744: 4740: 4736: 4733: 4730: 4727: 4724: 4718: 4713: 4709: 4699: 4693: 4688: 4684: 4679: 4673: 4668: 4659: 4653: 4649:". For every 4647: 4641: 4635: 4629: 4623: 4619: 4615: 4609: 4603: 4597: 4591: 4582: 4580: 4579:-invariants. 4577: 4572: 4568: 4563: 4557: 4552: 4547: 4542: 4539: 4532: 4526: 4521: 4517: 4511: 4505: 4499: 4495: 4489: 4485: 4481: 4475: 4471: 4465: 4460: 4456: 4450: 4448: 4443: 4437: 4431: 4426: 4421: 4415: 4409: 4406: 4400: 4394: 4388: 4382: 4376: 4370: 4366: 4362: 4356: 4352: 4347: 4343:. The subset 4341: 4336: 4331: 4325: 4319: 4315: 4311: 4305: 4301: 4297: 4291: 4287: 4282: 4278:. The subset 4275: 4271: 4267: 4263: 4259: 4255: 4248: 4244: 4238: 4233: 4228: 4219: 4217: 4215: 4211: 4205: 4201: 4198: 4191: 4187: 4182: 4174: 4166: 4157: 4153: 4147: 4143: 4137: 4131: 4125: 4122: 4116: 4110: 4104: 4101: 4097: 4093: 4087: 4083: 4079: 4073: 4067: 4062: 4057: 4054: 4050: 4046: 4042: 4036: 4030: 4025: 4020: 4016: 4012: 4006: 4000: 3995: 3991: 3987: 3982: 3977: 3972: 3967: 3961: 3955: 3949: 3936: 3930: 3927: 3924: 3921: 3918: 3914: 3910: 3904: 3901: 3897: 3893: 3884: 3880: 3874: 3868: 3862: 3856: 3850: 3844: 3839: 3830: 3824: 3814: 3808: 3802: 3798: 3792: 3788: 3783: 3778: 3773: 3766: 3764: 3762: 3758: 3757: 3751: 3746: 3742: 3738: 3733: 3726: 3718: 3716: 3714: 3710: 3705: 3701: 3697: 3691: 3687: 3682: 3681:smooth points 3678: 3673: 3668: 3663: 3657: 3654: 3650: 3644: 3640: 3636: 3631: 3626: 3621: 3617: 3613: 3609: 3603: 3599: 3593: 3589: 3585: 3578: 3574: 3568: 3563: 3558: 3556: 3551: 3544: 3540: 3536: 3530: 3526: 3519: 3515: 3510: 3504: 3500: 3496: 3492: 3488: 3481: 3477: 3473: 3469: 3464: 3455: 3453: 3448: 3444: 3440: 3435: 3430: 3425: 3420: 3413: 3405: 3403: 3400: 3396: 3392: 3387: 3383: 3379: 3373: 3369: 3364: 3363: 3357: 3353: 3348: 3342: 3340: 3336: 3329: 3325: 3321: 3317: 3310: 3306: 3302: 3296: 3290: 3286: 3281: 3277: 3273: 3268: 3264: 3260: 3256: 3252: 3248: 3241: 3235: 3228: 3224: 3220: 3214: 3210: 3204: 3200: 3195: 3192:if for every 3191: 3186: 3184: 3183: 3182:wandering set 3173: 3167: 3160: 3155: 3149: 3145: 3141: 3135: 3131: 3125: 3121: 3115: 3109: 3105: 3099: 3095: 3091: 3087: 3081: 3077: 3071: 3067: 3066:neighbourhood 3062: 3058: 3053: 3048: 3046: 3041: 3036: 3031: 3022: 3020: 3018: 3014: 3009: 3003: 2998: 2994: 2989: 2983: 2976: 2968: 2966: 2964: 2960: 2955: 2950: 2944: 2938: 2933: 2928: 2924: 2916: 2910: 2905: 2899: 2893: 2887: 2881: 2875: 2866: 2864: 2861: 2854: 2845:An action is 2843: 2841: 2837: 2831: 2825: 2819: 2815: 2808: 2804: 2799: 2795: 2791: 2785: 2781: 2775: 2771: 2764: 2760: 2755: 2751: 2744: 2740: 2735: 2731: 2725: 2720: 2716: 2709: 2704: 2700: 2693: 2686: 2680: 2676:has at least 2674: 2667: 2656: 2650: 2647: 2641: 2636: 2631: 2625: 2619: 2613: 2609: 2605: 2600: 2592: 2588: 2579: 2576: 2572: 2568: 2562: 2558: 2552: 2548: 2544: 2539: 2530: 2524: 2515: 2513: 2510: 2506: 2500: 2496: 2478: 2471: 2467: 2463: 2458: 2446: 2442: 2435: 2431: 2426: 2423: 2417: 2410: 2406: 2402: 2396: 2392: 2386: 2382: 2378: 2373: 2369: 2365: 2356: 2354: 2349: 2343: 2338: 2332: 2328: 2324: 2320:implies that 2318: 2314: 2308: 2304: 2300: 2295: 2287: 2278: 2272: 2263: 2261: 2259: 2254: 2247: 2243: 2230: 2227: 2223: 2196: 2190: 2146: 2143: 2136: 2132: 2129: 2126: 2122: 2115: 2111: 2107: 2097: 2094: 2093: 2078: 2075: 2072: 2068: 2064: 2057: 2054: 2053: 2050: 2049: 2048: 2045: 2041: 2035: 2028: 2024: 2020: 1993: 1990: 1987: 1984: 1978: 1975: 1969: 1966: 1960: 1957: 1954: 1948: 1942: 1935: 1932: 1931: 1916: 1913: 1907: 1904: 1901: 1895: 1888: 1885: 1884: 1881: 1880: 1879: 1862: 1859: 1853: 1850: 1847: 1844: 1841: 1834: 1833: 1832: 1822: 1814: 1812: 1804: 1785: 1780: 1775: 1771: 1735: 1731: 1724: 1721: 1715: 1709: 1705: 1701: 1694: 1690: 1683: 1669: 1666: 1663: 1659: 1655: 1648: 1647: 1646: 1643: 1637: 1633: 1626: 1622: 1618: 1612: 1606: 1602: 1598: 1593: 1589: 1584: 1580: 1575: 1571: 1545: 1537: 1534: 1530: 1526: 1520: 1509: 1505: 1501: 1496: 1492: 1485: 1476: 1468: 1464: 1455: 1451: 1443: 1442: 1441: 1424: 1421: 1415: 1407: 1403: 1395: 1394: 1393: 1390: 1386: 1379: 1375: 1369: 1365: 1360: 1356: 1352: 1347: 1342: 1337: 1335: 1327: 1324:is called a ( 1306: 1262: 1259: 1256: 1253: 1247: 1244: 1235: 1232: 1229: 1223: 1220: 1217: 1211: 1204: 1201: 1200: 1185: 1182: 1176: 1173: 1170: 1164: 1157: 1154: 1153: 1150: 1149: 1148: 1146: 1127: 1124: 1118: 1115: 1112: 1109: 1106: 1099: 1098: 1097: 1096: 1081: 1077: 1066: 1062: 1050: 1045: 1043: 1041: 1037: 1032: 1027: 1021: 1017: 1013: 1008: 1005: 1001: 996: 992: 988: 982: 978: 973: 969: 965: 961: 956: 954: 950: 946: 942: 938: 934: 930: 926: 921: 918: 913: 908: 903: 898: 894: 889: 884: 879: 877: 873: 868: 864: 860: 856: 852: 848: 844: 836: 826: 821: 810: 805: 803: 798: 796: 791: 790: 788: 787: 780: 777: 776: 773: 770: 769: 766: 763: 762: 759: 756: 755: 752: 747: 746: 736: 733: 730: 729: 727: 721: 718: 716: 713: 712: 709: 706: 704: 701: 699: 696: 695: 692: 686: 684: 678: 676: 670: 668: 662: 660: 654: 653: 649: 645: 642: 641: 637: 633: 630: 629: 625: 621: 618: 617: 613: 609: 606: 605: 601: 597: 594: 593: 589: 585: 582: 581: 577: 573: 570: 569: 565: 561: 558: 557: 554: 551: 549: 546: 545: 542: 538: 533: 532: 525: 522: 520: 517: 515: 512: 511: 483: 458: 457: 455: 449: 446: 421: 418: 417: 411: 408: 406: 403: 402: 398: 397: 386: 383: 381: 378: 375: 372: 371: 370: 369: 366: 362: 357: 354: 353: 350: 347: 346: 343: 340: 338: 336: 332: 331: 328: 325: 323: 320: 319: 316: 313: 311: 308: 307: 306: 305: 299: 296: 293: 288: 285: 284: 280: 275: 272: 269: 264: 261: 258: 253: 250: 249: 248: 247: 242: 241:Finite groups 237: 236: 225: 222: 220: 217: 216: 215: 214: 209: 206: 204: 201: 199: 196: 194: 191: 189: 186: 184: 181: 179: 176: 174: 171: 169: 166: 164: 161: 159: 156: 155: 154: 153: 148: 145: 143: 140: 139: 138: 137: 134: 133: 128: 123: 120: 118: 115: 113: 110: 108: 105: 102: 100: 97: 96: 95: 94: 89: 86: 84: 81: 79: 76: 75: 74: 73: 68:Basic notions 65: 64: 60: 56: 55: 52: 47: 43: 38: 30: 19: 8803:Group theory 8782: 8760: 8715: 8698: 8688:, retrieved 8684:the original 8678: 8651: 8635: 8609: 8590: 8571: 8567: 8539: 8516: 8497: 8470: 8445: 8433: 8421: 8402: 8396: 8386: 8379: 8360: 8354: 8346: 8341: 8329:. Retrieved 8314: 8307: 8295: 8283: 8276:Hatcher 2002 8271: 8259: 8247: 8235: 8223:. Retrieved 8219: 8210: 8200: 8192: 8182: 8175: 8058: 8044: 8034: 8024: 8010: 8007: 7997: 7991: 7981: 7975: 7972: 7961: 7934: 7931: 7924: 7918: 7912: 7906: 7900: 7894: 7885: 7881: 7875: 7869: 7865: 7859: 7853: 7847: 7843: 7837: 7828: 7822: 7816: 7809: 7803: 7798: 7793: 7787: 7781: 7774: 7769: 7763: 7760: 7755: 7750: 7743: 7738: 7732: 7726: 7720: 7714: 7707: 7703: 7699: 7695: 7691: 7687: 7681: 7677: 7673: 7667: 7661: 7656: 7651: 7645: 7639: 7636: 7630: 7621: 7617: 7613: 7606: 7604: 7586: 7576: 7572: 7566: 7560: 7554: 7546: 7541: 7536: 7532: 7528: 7522: 7516: 7512: 7508: 7504: 7498: 7492: 7486: 7480: 7474: 7468: 7447: 7441: 7435: 7429: 7426: 7423: 7419: 7415: 7408: 7404: 7400: 7396: 7390: 7361: 7355: 7349: 7343: 7336: 7332: 7328: 7324: 7320: 7316: 7310: 7300: 7294: 7288: 7278: 7274: 7270: 7264: 7260: 7256: 7250: 7246: 7242: 7238: 7232: 7228: 7224: 7217: 7213: 7207: 7203: 7199: 7192: 7188: 7184: 7177: 7173: 7169: 7161: 7155:is negative. 7151: 7145: 7138: 7132: 7126: 7122: 7116: 7110: 7104: 7098: 7082:well-behaved 7076:acts on the 7071: 7067:real numbers 7058: 7052: 7046: 7040: 7033: 7029: 7022: 7016: 7010: 7006: 6999:Galois group 6990: 6984: 6977: 6973: 6969: 6965: 6961: 6957: 6951: 6947: 6943: 6937: 6927: 6921: 6911: 6890: 6886:Möbius group 6875: 6867: 6863: 6853: 6849: 6838:affine space 6833: 6830:affine space 6826:transitively 6822:affine group 6813: 6803: 6796: 6792: 6782: 6776: 6765: 6761: 6750: 6746: 6738: 6734: 6723: 6719: 6707: 6703: 6680: 6676: 6669: 6662: 6657: 6653: 6646: 6639: 6635: 6629: 6625: 6621: 6614: 6608: 6602: 6579: 6571: 6560: 6550: 6544: 6540: 6534: 6530:automorphism 6524: 6518: 6509: 6503: 6497: 6491: 6485: 6481: 6477: 6471: 6465: 6459: 6453: 6444: 6440: 6434: 6431: 6427: 6421: 6417: 6413: 6407: 6401: 6391: 6378: 6372: 6362: 6356: 6350: 6344: 6338: 6334: 6330: 6324: 6320: 6314: 6308: 6302: 6293: 6283: 6277: 6271: 6265: 6261: 6257: 6251: 6245: 6239: 6230: 6220: 6214: 6208: 6202: 6196: 6192: 6188: 6182: 6176: 6167: 6145: 6135: 6129: 6126: 6120: 6114: 6108: 6102: 5993: 5973: 5956: 5939: 5930: 5911: 5896: 5882: 5867: 5857: 5850: 5835: 5824: 5814: 5803: 5792: 5783: 5776: 5770: 5758: 5756:group. Then 5754:automorphism 5748: 5735: 5719: 5716: 5709: 5703: 5697: 5691: 5681: 5677: 5671: 5661: 5655: 5649: 5643: 5638: 5626: 5509: 5506: 5501: 5496: 5492: 5487: 5483: 5477: 5473: 5466: 5462: 5458: 5448: 5442: 5438: 5432: 5426: 5419: 5415: 5410:. Thus, the 5405: 5401: 5391: 5385: 5381: 5376: 5372: 5368: 5364: 5188: 5184: 5177: 5173: 5167: 5163: 5159: 5153: 5149: 5145: 5139: 5133: 5130: 5110: 5103: 5097: 5095:of some/any 5090: 5086: 5079: 5072: 5066: 5059: 5052: 5041: 5035: 5031: 5027: 5020: 5016: 5012: 5005: 5001: 4997: 4994: 4988: 4984: 4980: 4977: 4970: 4964: 4960: 4956: 4952: 4948: 4941: 4937: 4933: 4927: 4923: 4919: 4914: 4910: 4903: 4899: 4892: 4888: 4882: 4878: 4874: 4868: 4862: 4856: 4850: 4847: 4841: 4835: 4829: 4822: 4818: 4814:intersection 4807: 4803: 4797: 4787: 4781: 4775: 4697: 4691: 4687:little group 4686: 4682: 4677: 4671: 4662: 4657: 4651: 4645: 4639: 4633: 4627: 4621: 4617: 4613: 4607: 4601: 4595: 4589: 4586: 4575: 4561: 4555: 4545: 4537: 4530: 4524: 4519: 4515: 4509: 4503: 4497: 4493: 4487: 4483: 4479: 4473: 4469: 4463: 4458: 4454: 4451: 4446: 4441: 4435: 4429: 4425:transitively 4419: 4413: 4410: 4404: 4398: 4392: 4386: 4380: 4374: 4368: 4364: 4360: 4354: 4351:fixed under 4350: 4345: 4339: 4329: 4323: 4317: 4313: 4309: 4303: 4299: 4295: 4289: 4285: 4280: 4273: 4269: 4265: 4261: 4257: 4253: 4246: 4242: 4236: 4226: 4223: 4196: 4189: 4185: 4179:coinvariants 4176: 4168: 4160: 4155: 4151: 4145: 4141: 4135: 4129: 4126: 4120: 4118:(given that 4114: 4108: 4099: 4095: 4091: 4085: 4081: 4077: 4071: 4065: 4058: 4052: 4048: 4044: 4040: 4034: 4028: 4018: 4014: 4010: 4004: 3998: 3989: 3985: 3975: 3965: 3959: 3953: 3950: 3882: 3878: 3872: 3866: 3860: 3854: 3848: 3842: 3833: 3828: 3822: 3819: 3812: 3806: 3800: 3796: 3790: 3781: 3754: 3749: 3731: 3728: 3703: 3699: 3695: 3689: 3685: 3680: 3671: 3661: 3658: 3652: 3648: 3642: 3638: 3634: 3629: 3627: 3619: 3615: 3611: 3607: 3601: 3597: 3591: 3587: 3583: 3576: 3572: 3566: 3562:locally free 3561: 3559: 3549: 3542: 3538: 3534: 3528: 3524: 3517: 3513: 3502: 3498: 3494: 3490: 3486: 3479: 3475: 3471: 3467: 3458: 3456: 3446: 3442: 3438: 3433: 3428: 3418: 3415: 3398: 3394: 3385: 3381: 3377: 3371: 3367: 3360: 3355: 3346: 3343: 3338: 3334: 3327: 3323: 3319: 3315: 3308: 3304: 3300: 3294: 3288: 3284: 3269: 3262: 3258: 3254: 3250: 3246: 3239: 3233: 3226: 3222: 3218: 3212: 3208: 3202: 3198: 3189: 3187: 3180: 3171: 3165: 3158: 3153: 3147: 3143: 3139: 3133: 3129: 3123: 3119: 3113: 3107: 3103: 3100: 3093: 3089: 3085: 3079: 3075: 3069: 3060: 3056: 3051: 3049: 3039: 3029: 3027:Assume that 3026: 3007: 3001: 2992: 2987: 2981: 2978: 2953: 2942: 2936: 2929: 2922: 2914: 2903: 2897: 2891: 2885: 2879: 2873: 2870: 2859: 2848: 2844: 2829: 2823: 2817: 2813: 2806: 2802: 2797: 2793: 2789: 2783: 2779: 2773: 2769: 2762: 2758: 2753: 2749: 2742: 2738: 2733: 2729: 2723: 2718: 2714: 2707: 2702: 2698: 2691: 2684: 2678: 2672: 2662: 2654: 2651: 2645: 2639: 2629: 2623: 2617: 2615:the element 2611: 2607: 2603: 2594: 2590: 2582: 2580: 2574: 2570: 2566: 2560: 2556: 2550: 2546: 2542: 2533: 2528: 2522: 2519: 2508: 2504: 2498: 2494: 2479: 2469: 2465: 2462:cyclic group 2456: 2444: 2440: 2427: 2421: 2415: 2408: 2404: 2400: 2394: 2390: 2384: 2380: 2376: 2371: 2367: 2359: 2357: 2347: 2341: 2337:homomorphism 2330: 2326: 2322: 2316: 2312: 2306: 2302: 2298: 2289: 2281: 2276: 2270: 2267: 2255: 2245: 2228: 2225: 2221: 2194: 2191: 2168: 2043: 2039: 2033: 2026: 2022: 2018: 2015: 1877: 1820: 1819:Likewise, a 1818: 1802: 1783: 1773: 1769: 1750: 1641: 1635: 1631: 1624: 1620: 1616: 1610: 1607: 1600: 1596: 1591: 1587: 1582: 1578: 1563: 1439: 1388: 1384: 1377: 1373: 1367: 1363: 1358: 1354: 1345: 1338: 1333: 1325: 1307: 1284: 1142: 1080:group action 1079: 1075: 1054: 1036:permutations 1030: 1024:acts on any 1019: 1015: 1009: 1003: 994: 980: 976: 962:is called a 960:vector space 957: 943:acts on the 922: 916: 906: 896: 887: 883:group action 882: 881:Formally, a 880: 875: 867:group action 866: 840: 825:cyclic group 647: 635: 623: 611: 599: 587: 575: 563: 334: 291: 278: 267: 256: 252:Cyclic group 130: 117:Free product 88:Group action 87: 51:Group theory 46:Group theory 45: 8331:23 February 8288:Maskit 1988 8225:19 December 7835:Every free 7797:are called 7776:isomorphism 7466:Given left 7378:quaternions 7078:phase space 6882:cross ratio 6397:conjugation 6180:on any set 5949:) ⋅ 3 5906:) ⋅ 3 5873:) ⋅ 2 5830:) ⋅ 2 5752:denote its 5107:belongs to 4992:; that is, 4968:. Applying 4637:" or that " 4520:-invariants 4507:is denoted 4461:element of 4335:restricting 4171:orbit space 3745:irreducible 3522:the set of 3483:defined by 3465:if the map 3436:if the map 3416:Now assume 2963:unit sphere 2945:∖ {0} 2940:on the set 2853:-transitive 2666:-transitive 2368:semiregular 1811:to itself. 1343:the action 1040:cardinality 920:to itself. 885:of a group 843:mathematics 537:Topological 376:alternating 8797:Categories 8707:0873.57001 8690:2016-02-08 8644:0627.30039 8628:1180.57001 8460:References 8345:M. Artin, 8220:Proof Wiki 8136:Gain graph 8016:invertible 7799:isomorphic 7685:such that 7353:and every 7268:, but not 6982:for every 6955:such that 6900:isometries 6772:Lie groups 6556:involution 5951:| = 2 5875:| = 3 5689:which are 5452:induces a 4866:, and let 4769:This is a 4551:cohomology 4477:such that 4459:-invariant 4447:transitive 4349:is called 4061:transitive 3761:direct sum 3756:semisimple 3581:such that 3532:such that 3434:continuous 3375:such that 3359:is called 3313:for every 3298:such that 3216:such that 3017:singletons 2991:is called 2816:= 1, ..., 2787:such that 2649:-torsor. 2536:transitive 2532:is called 2055:Identity: 1886:Identity: 1308:The group 1155:Identity: 1046:Definition 951:, and the 941:polyhedron 937:symmetries 644:Symplectic 584:Orthogonal 541:Lie groups 448:Free group 173:continuous 112:Direct sum 8784:MathWorld 8767:EMS Press 8167:Citations 7948:metalogic 7655:-sets, a 7306:power set 6933:morphisms 6633:given by 6578:{1, ..., 6522:on a set 6060:∈ 6053:∑ 5779:⋅ 1 5533:⋅ 5454:bijection 5430:over any 5334:∈ 5327:⟺ 5313:∈ 5302:− 5293:⟺ 5279:⋅ 5267:− 5258:⟺ 5250:⋅ 5236:⋅ 5228:⟺ 5157:given by 5076:has type 5044:conjugate 4741:⋅ 4728:∈ 4695:that fix 4553:group of 4417:on which 3971:partition 3928:∈ 3915:⋅ 3898:⋅ 3858:to which 3667:Lie group 3614:∖ { 3362:cocompact 3322:∖ { 3244:given by 3054:if every 3052:wandering 2997:partition 2993:primitive 2388:for some 2353:injective 2292:effective 2137:⋅ 2123:⋅ 2112:⋅ 2069:⋅ 1979:α 1949:α 1943:α 1896:α 1857:→ 1851:× 1845:: 1842:α 1779:bijection 1732:⋅ 1706:⋅ 1695:⋅ 1660:⋅ 1531:α 1506:α 1502:∘ 1493:α 1465:α 1452:α 1404:α 1248:α 1224:α 1212:α 1165:α 1122:→ 1116:× 1110:: 1107:α 991:dimension 968:subgroups 933:triangles 872:structure 859:rotations 835:rotations 708:Conformal 596:Euclidean 203:nilpotent 8818:Symmetry 8537:(2002), 8469:(2000). 8278:, p. 72. 8254:, p. 73. 8130:See also 8054:groupoid 7940:category 7863:acts on 7851:, where 7724:and all 7712:for all 7676: : 7657:morphism 7649:are two 7609:groupoid 7564:). This 7551:(where " 7507: : 7331: : 6946: : 6917:category 6801:acts on 6489:for all 6342:for all 6269:for all 6212:and all 6200:for all 6161:Examples 5736:Example: 5639:Example: 5517:, gives 5148: : 4983:)⋅ 4951:⋅( 4827:for all 4771:subgroup 4491:for all 4384:and all 4372:for all 4260: : 4202:quotient 4163:quotient 3595:for all 3384:⋅ 3249:⋅( 2867:Examples 2849:sharply 2688:-tuples 2569:⋅ 2564:so that 2434:embedded 2310:for all 2284:faithful 2198:acts on 2181:and all 2169:for all 1362: : 1297:and all 1285:for all 1095:function 945:vertices 927:acts on 703:Poincaré 548:Solenoid 420:Integers 410:Lattices 385:sporadic 380:Lie type 208:solvable 198:dihedral 183:additive 168:infinite 78:Subgroup 8769:, 2001 8743:0889050 8559:1867354 8489:1777008 8347:Algebra 8099:Gallery 8089:schemes 8081:actions 8038:to the 8028:to the 8020:functor 7964:monoids 7950:, this 7607:action 7386:versors 7384:1 (the 7304:on the 7144:− 6889:PGL(2, 6874:PGL(2, 6834:regular 6675:, ..., 6652:, ..., 6170:trivial 5495:⋅ 5476:⋅ 5441:⋅ 5187:⋅ 5166:⋅ 5034:⋅ 4963:⋅ 4955:⋅ 4881:⋅ 4616:⋅ 4571:functor 4569:of the 4541:-module 4528:. When 4482:⋅ 4363:⋅ 4312:⋅ 4298:⋅ 4256:⋅ 4245:⋅ 4240:, then 4094:⋅ 4080:⋅ 4051:⋅ 4043:⋅ 4013:⋅ 3969:form a 3881:⋅ 3775:In the 3739:over a 3702:⋅ 3641:⋅ 3586:⋅ 3537:⋅ 3501:⋅ 3303:⋅ 3274:of the 3221:⋅ 3196:subset 3194:compact 3142:⋅ 3088:⋅ 2792:⋅ 2713:, ..., 2697:, ..., 2597:regular 2379:⋅ 2258:induces 2042:⋅ 1772:⋅ 1634:⋅ 998:over a 970:of the 849:form a 698:Lorentz 620:Unitary 519:Lattice 459:PSL(2, 193:abelian 104:(Semi-) 8741: 8731: 8705: 8666: 8642: 8626: 8616: 8597: 8578: 8557: 8547: 8523: 8504: 8487: 8477: 8409: 8367: 8322: 8079:, and 7928:-set.) 7785:-sets 7590:-sets. 7537:α 7533:α 7505:α 7472:-sets 7399:= cos 7102:is in 6884:; the 6770:) are 6755:, and 6597:For a 6384:degree 6100:where 5979:| 5936:| 5926:| 5879:| 5864:| 5858:π 5846:| 5812:| 5799:| 5768:| 4806:→ Sym( 4661:, the 4643:fixes 4587:Given 4232:subset 4206:subset 3832:. The 3737:module 3709:smooth 3509:proper 3461:proper 3257:) = (2 3152:. The 3064:has a 3043:is by 2832:= 2, 3 2019:α 2016:(with 1617:α 1611:α 1597:α 1588:α 1579:α 1568:being 1374:α 1355:α 1346:α 1145:axioms 1083:α 1070:, and 947:, the 853:under 553:Circle 484:SL(2, 373:cyclic 337:-group 188:cyclic 163:finite 158:simple 142:kernel 8162:Notes 8022:from 7995:. If 7952:topos 7659:from 7633:-sets 7539:∘ (id 7490:-set 7403:/2 + 7380:with 7254:, or 7096:: if 7092:) by 7001:of a 6968:)) = 6852:+ 1, 6824:acts 6695:graph 6661:) ↦ ( 5740:graph 5659:with 5412:fiber 5397:coset 4839:. If 4611:with 4534:is a 4423:acts 4230:is a 4075:with 4008:with 3836:orbit 3665:is a 3553:is a 3545:′ ≠ ∅ 3493:) ↦ ( 3422:is a 3350:on a 3137:with 3083:with 3033:is a 2895:. If 2767:when 2643:or a 2593:, or 2507:/ 120 2339:from 1777:is a 1564:with 1341:curry 1093:is a 1063:with 1061:group 1059:is a 1028:with 1000:field 953:faces 949:edges 939:of a 904:from 900:is a 891:on a 851:group 737:Sp(∞) 734:SU(∞) 147:image 8729:ISBN 8664:ISBN 8614:ISBN 8595:ISBN 8576:ISBN 8545:ISBN 8521:ISBN 8502:ISBN 8475:ISBN 8407:ISBN 8365:ISBN 8333:2017 8320:ISBN 8227:2021 7791:and 7756:maps 7698:) = 7643:and 7616:′ = 7422:) = 7407:sin 7382:norm 7376:The 7263:) + 7206:) + 7164:, +) 7108:and 7080:of " 7074:, +) 7056:and 7028:Gal( 6997:The 6935:are 6898:The 6848:PGL( 6843:The 6820:The 6624:* × 6587:The 6501:and 6443:) = 6166:The 6118:and 5680:mod 5641:Let 5389:and 5371:) = 5025:and 4959:) = 4897:and 4854:and 4848:Let 4793:free 4599:and 4268:and 4212:and 4103:for 4032:and 3957:in) 3669:and 3605:and 3426:and 3164:Ω ⊂ 3019:). 3013:dual 2925:− 1) 2917:− 2) 2811:for 2722:) ∈ 2706:), ( 2637:for 2589:(or 2366:(or 2362:free 2268:Let 2224:) = 2173:and 1801:Sym( 1440:and 1326:left 1289:and 1076:left 1010:The 876:acts 823:The 731:O(∞) 720:Loop 539:and 8721:doi 8703:Zbl 8656:doi 8640:Zbl 8624:Zbl 8572:148 8087:on 8083:of 8075:on 7989:of 7916:. ( 7910:of 7826:on 7748:or 7730:in 7718:in 7665:to 7637:If 7545:× – 7359:in 7347:of 7323:= { 7308:of 7292:on 6988:in 6919:of 6791:GL( 6760:Sp( 6745:SO( 6718:SL( 6702:GL( 6638:× ( 6558:of 6548:on 6543:/ 2 6532:of 6495:in 6486:gKg 6422:gxg 6405:on 6370:of 6354:in 6339:gaH 6281:in 6249:on 6228:on 6218:in 6206:in 6143:of 5669:or 5507:If 5436:in 5424:of 5137:in 5101:in 4833:in 4791:is 4785:on 4773:of 4685:or 4669:of 4655:in 4605:in 4593:in 4573:of 4522:of 4467:is 4445:is 4439:on 4390:in 4378:in 4358:if 4333:by 4293:if 4234:of 4224:If 4112:in 4105:all 4069:in 4002:in 3973:of 3846:in 3729:If 3707:is 3659:If 3625:. 3570:of 3557:. 3507:is 3341:. 3311:= ∅ 3261:, 2 3237:on 3229:≠ ∅ 3185:. 3175:on 3150:≠ ∅ 3096:≠ ∅ 3047:. 2999:of 2985:on 2911:is 2670:if 2657:≥ 1 2578:. 2526:on 2497:/ 2 2468:/ 2 2443:/ 2 2370:or 2296:if 2288:or 2249:on 2236:on 2185:in 2177:in 2037:or 1827:on 1823:of 1791:on 1767:to 1755:in 1639:or 1334:set 1301:in 1293:in 1089:on 1085:of 1055:If 1026:set 989:of 975:GL( 893:set 841:In 646:Sp( 634:SU( 610:SO( 574:SL( 562:GL( 8799:: 8781:. 8765:, 8759:, 8739:MR 8737:, 8727:, 8662:, 8622:, 8555:MR 8553:, 8485:MR 8483:. 8218:. 7970:. 7889:.) 7884:/ 7868:× 7846:× 7758:. 7680:→ 7620:⋉ 7575:→ 7535:= 7515:→ 7511:× 7478:, 7457:−1 7445:; 7411:/2 7335:∈ 7277:+ 7275:xe 7261:xe 7245:+ 7236:, 7222:, 7218:xe 7211:, 7197:, 7191:+ 7176:)( 7125:+ 7032:/ 7009:/ 6960:⋅( 6950:→ 6795:, 6764:, 6749:, 6737:, 6733:O( 6728:, 6722:, 6706:, 6677:ax 6670:ax 6668:, 6663:ax 6645:, 6628:→ 6484:= 6475:: 6435:xg 6430:= 6420:= 6411:: 6395:, 6348:, 6337:= 6335:aH 6328:: 6323:/ 6275:, 6266:gx 6264:= 6255:: 6195:= 6157:. 5998:: 5955:(( 5938:(( 5910:(( 5895:(( 5860:/3 5504:. 5491:↦ 5484:gG 5461:/ 5422:}) 5418:({ 5162:↦ 5152:→ 5119:. 5039:. 5030:= 5015:∈ 5000:∈ 4998:hg 4987:= 4981:hg 4936:∈ 4920:gG 4918:= 4877:= 4701:: 4620:= 4543:, 4496:∈ 4486:= 4472:∈ 4452:A 4367:= 4316:⊆ 4302:= 4272:∈ 4264:∈ 4154:\ 4144:/ 4098:= 4084:= 4056:. 4047:= 4017:= 3988:~ 3886:: 3807:gT 3799:/ 3698:↦ 3688:∈ 3675:a 3651:∈ 3637:↦ 3610:∈ 3600:∈ 3590:≠ 3541:∩ 3527:∈ 3516:, 3497:, 3489:, 3478:× 3474:→ 3470:× 3454:. 3445:→ 3441:× 3402:. 3397:\ 3380:= 3370:⊂ 3318:∈ 3307:∩ 3287:∈ 3253:, 3225:∩ 3211:∈ 3201:⊂ 3146:∩ 3132:∈ 3122:∋ 3106:∈ 3098:. 3092:∩ 3078:∈ 3059:∈ 2965:. 2801:= 2782:∈ 2772:≠ 2757:≠ 2747:, 2737:≠ 2610:∈ 2606:, 2573:= 2559:∈ 2549:∈ 2545:, 2493:× 2477:. 2403:= 2393:∈ 2383:= 2355:. 2325:= 2315:∈ 2305:= 2253:. 2222:gh 2195:gh 2189:. 2034:xg 2025:, 1642:gx 1623:, 1605:. 1601:gh 1595:= 1586:∘ 1387:∈ 1366:→ 1336:. 1328:) 1305:. 1147:: 1078:) 1042:. 1007:. 979:, 622:U( 598:E( 586:O( 44:→ 8787:. 8723:: 8658:: 8603:. 8584:. 8562:. 8529:. 8510:. 8491:. 8415:. 8373:. 8335:. 8302:. 8229:. 8045:G 8035:G 8025:G 8011:G 7998:X 7992:X 7982:X 7976:X 7935:G 7925:G 7919:H 7913:G 7907:H 7901:G 7895:G 7886:G 7882:X 7876:S 7870:S 7866:G 7860:G 7854:S 7848:S 7844:G 7838:G 7829:G 7823:G 7817:G 7804:G 7794:Y 7788:X 7782:G 7770:f 7764:f 7754:- 7751:G 7739:G 7733:X 7727:x 7721:G 7715:g 7710:) 7708:x 7706:( 7704:f 7702:⋅ 7700:g 7696:x 7694:⋅ 7692:g 7690:( 7688:f 7682:Y 7678:X 7674:f 7668:Y 7662:X 7652:G 7646:Y 7640:X 7631:G 7622:X 7618:G 7614:G 7587:G 7577:Y 7573:X 7567:G 7561:g 7555:g 7553:– 7549:) 7547:g 7542:X 7531:⋅ 7529:g 7523:G 7517:Y 7513:G 7509:X 7499:G 7493:Y 7487:G 7481:Y 7475:X 7469:G 7463:. 7461:1 7448:z 7442:v 7436:α 7430:z 7427:x 7424:z 7420:x 7418:( 7416:f 7409:α 7405:v 7401:α 7397:z 7391:R 7373:. 7362:G 7356:g 7350:X 7344:U 7339:} 7337:U 7333:u 7329:u 7327:⋅ 7325:g 7321:U 7319:⋅ 7317:g 7311:X 7301:G 7295:X 7289:G 7283:. 7281:) 7279:t 7273:( 7271:f 7265:t 7259:( 7257:f 7251:e 7249:) 7247:t 7243:x 7241:( 7239:f 7233:e 7231:) 7229:x 7227:( 7225:f 7220:) 7216:( 7214:f 7208:t 7204:x 7202:( 7200:f 7195:) 7193:t 7189:x 7187:( 7185:f 7180:) 7178:x 7174:f 7172:⋅ 7170:t 7168:( 7162:R 7160:( 7152:t 7146:t 7139:t 7133:t 7127:x 7123:t 7117:x 7111:x 7105:R 7099:t 7072:R 7070:( 7062:. 7059:K 7053:L 7047:K 7041:L 7036:) 7034:K 7030:L 7023:K 7017:L 7011:K 7007:L 6994:. 6991:G 6985:g 6980:) 6978:x 6976:⋅ 6974:g 6972:( 6970:f 6966:x 6964:( 6962:f 6958:g 6952:Y 6948:X 6944:f 6938:G 6928:G 6922:G 6912:G 6893:) 6891:C 6878:) 6876:K 6870:) 6868:K 6866:( 6864:P 6856:) 6854:K 6850:n 6840:. 6817:. 6814:Z 6804:Z 6799:) 6797:Z 6793:n 6786:. 6783:K 6777:K 6768:) 6766:K 6762:n 6753:) 6751:K 6747:n 6741:) 6739:K 6735:n 6726:) 6724:K 6720:n 6710:) 6708:K 6704:n 6690:. 6684:) 6681:n 6673:2 6666:1 6658:n 6654:x 6650:2 6647:x 6643:1 6640:x 6636:a 6630:V 6626:V 6622:F 6617:* 6615:F 6609:F 6603:V 6582:} 6580:n 6572:n 6569:S 6564:. 6561:X 6551:X 6545:Z 6541:Z 6535:X 6525:X 6519:Z 6513:. 6510:H 6504:K 6498:G 6492:g 6482:K 6480:⋅ 6478:g 6472:H 6466:G 6460:H 6454:G 6448:. 6445:x 6441:x 6432:g 6428:x 6418:x 6416:⋅ 6414:g 6408:G 6402:G 6392:G 6386:. 6379:G 6373:G 6363:H 6357:G 6351:a 6345:g 6333:⋅ 6331:g 6325:H 6321:G 6315:G 6309:H 6303:G 6297:. 6294:G 6284:G 6278:x 6272:g 6262:x 6260:⋅ 6258:g 6252:G 6246:G 6240:G 6234:. 6231:X 6221:X 6215:x 6209:G 6203:g 6197:x 6193:x 6191:⋅ 6189:g 6183:X 6177:G 6146:G 6136:G 6130:G 6121:X 6115:G 6109:g 6103:X 6088:, 6084:| 6078:g 6074:X 6069:| 6063:G 6057:g 6045:| 6041:G 6037:| 6032:1 6027:= 6023:| 6019:G 6015:/ 6011:X 6007:| 5985:. 5981:G 5974:G 5968:3 5966:) 5964:2 5962:) 5960:1 5957:G 5947:2 5945:) 5943:1 5940:G 5931:G 5923:3 5921:) 5919:2 5917:) 5915:1 5912:G 5904:2 5902:) 5900:1 5897:G 5890:2 5888:) 5886:1 5883:G 5881:( 5871:1 5868:G 5866:( 5856:2 5851:G 5843:2 5841:) 5839:1 5836:G 5834:( 5828:1 5825:G 5823:( 5818:1 5815:G 5807:1 5804:G 5796:1 5793:G 5787:1 5784:G 5777:G 5771:G 5759:G 5749:G 5720:X 5713:. 5710:p 5704:G 5698:k 5692:G 5687:1 5682:p 5678:k 5672:p 5667:1 5662:k 5656:X 5650:p 5644:G 5627:x 5612:, 5608:| 5602:x 5598:G 5593:| 5588:/ 5583:| 5579:G 5575:| 5571:= 5568:] 5563:x 5559:G 5554:: 5550:G 5547:[ 5544:= 5540:| 5536:x 5530:G 5526:| 5510:G 5497:x 5493:g 5488:x 5478:x 5474:G 5467:x 5463:G 5459:G 5449:f 5443:x 5439:G 5433:y 5427:f 5420:y 5416:f 5406:x 5402:G 5392:h 5386:g 5379:) 5377:h 5375:( 5373:f 5369:g 5367:( 5365:f 5350:. 5345:x 5341:G 5337:g 5331:h 5321:x 5317:G 5310:h 5305:1 5298:g 5289:x 5286:= 5283:x 5275:h 5270:1 5263:g 5254:x 5246:h 5243:= 5240:x 5232:g 5224:) 5221:h 5218:( 5215:f 5212:= 5209:) 5206:g 5203:( 5200:f 5189:x 5185:G 5180:) 5178:G 5176:( 5174:f 5168:x 5164:g 5160:g 5154:X 5150:G 5146:f 5140:X 5134:x 5113:) 5111:H 5109:( 5104:O 5098:x 5091:x 5087:G 5082:) 5080:H 5078:( 5073:O 5067:H 5062:) 5060:H 5058:( 5053:G 5036:y 5032:g 5028:x 5021:x 5017:G 5013:h 5006:x 5002:G 4995:g 4989:x 4985:x 4978:g 4976:( 4971:g 4965:x 4961:g 4957:x 4953:g 4949:h 4942:y 4938:G 4934:h 4928:g 4924:x 4915:y 4911:G 4904:y 4900:G 4893:x 4889:G 4883:x 4879:g 4875:y 4869:g 4863:X 4857:y 4851:x 4842:N 4836:X 4830:x 4823:x 4819:G 4810:) 4808:X 4804:G 4798:N 4788:X 4782:G 4776:G 4757:. 4754:} 4751:x 4748:= 4745:x 4737:g 4734:: 4731:G 4725:g 4722:{ 4719:= 4714:x 4710:G 4698:x 4692:G 4678:x 4672:G 4658:X 4652:x 4646:x 4640:g 4634:g 4628:x 4622:x 4618:x 4614:g 4608:X 4602:x 4596:G 4590:g 4576:G 4562:X 4556:G 4546:X 4538:G 4531:X 4525:X 4516:G 4510:X 4504:x 4498:G 4494:g 4488:x 4484:x 4480:g 4474:X 4470:x 4464:X 4455:G 4442:X 4436:G 4430:X 4420:G 4414:X 4405:G 4399:G 4393:Y 4387:y 4381:G 4375:g 4369:y 4365:y 4361:g 4355:G 4346:Y 4340:Y 4330:Y 4324:G 4318:Y 4314:Y 4310:G 4304:Y 4300:Y 4296:G 4290:G 4281:Y 4276:} 4274:Y 4270:y 4266:G 4262:g 4258:y 4254:g 4252:{ 4247:Y 4243:G 4237:X 4227:Y 4197:X 4190:G 4186:X 4156:X 4152:G 4146:G 4142:X 4136:G 4130:X 4121:X 4115:X 4109:x 4100:X 4096:x 4092:G 4086:X 4082:x 4078:G 4072:X 4066:x 4053:y 4049:G 4045:x 4041:G 4035:y 4029:x 4019:y 4015:x 4011:g 4005:G 3999:g 3990:y 3986:x 3976:X 3966:G 3960:X 3954:x 3937:. 3934:} 3931:G 3925:g 3922:: 3919:x 3911:g 3908:{ 3905:= 3902:x 3894:G 3883:x 3879:G 3873:x 3867:G 3861:x 3855:X 3849:X 3843:x 3829:X 3823:G 3813:g 3801:T 3797:I 3791:T 3782:I 3750:g 3732:g 3704:x 3700:g 3696:g 3690:X 3686:x 3672:X 3662:G 3653:X 3649:x 3643:x 3639:g 3635:g 3623:} 3620:G 3616:e 3612:U 3608:g 3602:X 3598:x 3592:x 3588:x 3584:g 3577:G 3573:e 3567:U 3550:G 3543:K 3539:K 3535:g 3529:G 3525:g 3520:′ 3518:K 3514:K 3505:) 3503:x 3499:g 3495:x 3491:x 3487:g 3485:( 3480:X 3476:X 3472:X 3468:G 3447:X 3443:X 3439:G 3429:X 3419:G 3399:X 3395:G 3386:A 3382:G 3378:X 3372:X 3368:A 3356:X 3347:G 3331:} 3328:G 3324:e 3320:G 3316:g 3309:U 3305:U 3301:g 3295:U 3289:X 3285:x 3265:) 3263:y 3259:x 3255:y 3251:x 3247:n 3240:R 3234:Z 3227:K 3223:K 3219:g 3213:G 3209:g 3203:X 3199:K 3177:Ω 3172:G 3166:X 3159:G 3148:U 3144:U 3140:g 3134:G 3130:g 3124:x 3120:U 3114:G 3108:X 3104:x 3094:U 3090:U 3086:g 3080:G 3076:g 3070:U 3061:X 3057:x 3040:G 3030:X 3008:G 3002:X 2988:X 2982:G 2954:v 2943:V 2937:V 2923:n 2921:( 2915:n 2913:( 2904:n 2898:X 2892:X 2886:n 2880:n 2874:X 2860:X 2851:n 2830:n 2824:X 2818:n 2814:i 2807:i 2803:y 2798:i 2794:x 2790:g 2784:G 2780:g 2774:j 2770:i 2763:j 2759:y 2754:i 2750:y 2743:j 2739:x 2734:i 2730:x 2724:X 2719:n 2715:y 2711:1 2708:y 2703:n 2699:x 2695:1 2692:x 2690:( 2685:n 2679:n 2673:X 2664:n 2655:n 2646:G 2640:G 2630:G 2624:X 2618:g 2612:X 2608:y 2604:x 2575:y 2571:x 2567:g 2561:G 2557:g 2551:X 2547:y 2543:x 2529:X 2523:G 2509:Z 2505:Z 2499:Z 2495:Z 2491:5 2489:A 2484:5 2482:S 2475:2 2470:Z 2466:Z 2457:n 2455:2 2451:2 2447:) 2445:Z 2441:Z 2439:( 2422:X 2416:G 2409:G 2405:e 2401:g 2395:X 2391:x 2385:x 2381:x 2377:g 2348:X 2342:G 2331:G 2327:e 2323:g 2317:X 2313:x 2307:x 2303:x 2301:⋅ 2299:g 2277:X 2271:G 2251:X 2246:G 2238:X 2234:G 2229:g 2226:h 2220:( 2216:h 2212:g 2208:g 2204:h 2200:x 2187:X 2183:x 2179:G 2175:h 2171:g 2150:) 2147:h 2144:g 2141:( 2133:x 2130:= 2127:h 2119:) 2116:g 2108:x 2105:( 2079:x 2076:= 2073:e 2065:x 2044:g 2040:x 2029:) 2027:g 2023:x 2021:( 1997:) 1994:h 1991:g 1988:, 1985:x 1982:( 1976:= 1973:) 1970:h 1967:, 1964:) 1961:g 1958:, 1955:x 1952:( 1946:( 1917:x 1914:= 1911:) 1908:e 1905:, 1902:x 1899:( 1863:, 1860:X 1854:G 1848:X 1829:X 1825:G 1809:X 1805:) 1803:X 1797:G 1793:X 1789:G 1784:g 1774:x 1770:g 1765:x 1761:X 1757:G 1753:g 1736:x 1728:) 1725:h 1722:g 1719:( 1716:= 1713:) 1710:x 1702:h 1699:( 1691:g 1670:x 1667:= 1664:x 1656:e 1636:x 1632:g 1627:) 1625:x 1621:g 1619:( 1592:h 1583:g 1566:∘ 1549:) 1546:x 1543:( 1538:h 1535:g 1527:= 1524:) 1521:x 1518:( 1515:) 1510:h 1497:g 1489:( 1486:= 1483:) 1480:) 1477:x 1474:( 1469:h 1461:( 1456:g 1425:x 1422:= 1419:) 1416:x 1413:( 1408:e 1389:G 1385:g 1378:g 1368:X 1364:X 1359:g 1332:- 1330:G 1322:G 1318:X 1314:X 1310:G 1303:X 1299:x 1295:G 1291:h 1287:g 1266:) 1263:x 1260:, 1257:h 1254:g 1251:( 1245:= 1242:) 1239:) 1236:x 1233:, 1230:h 1227:( 1221:, 1218:g 1215:( 1186:x 1183:= 1180:) 1177:x 1174:, 1171:e 1168:( 1128:, 1125:X 1119:X 1113:G 1091:X 1087:G 1072:X 1068:e 1057:G 1031:n 1020:n 1016:S 1004:K 995:n 983:) 981:K 977:n 917:S 907:G 897:S 888:G 830:3 828:C 808:e 801:t 794:v 690:8 688:E 682:7 680:E 674:6 672:E 666:4 664:F 658:2 656:G 650:) 648:n 638:) 636:n 626:) 624:n 614:) 612:n 602:) 600:n 590:) 588:n 578:) 576:n 566:) 564:n 506:) 493:Z 481:) 468:Z 444:) 431:Z 422:( 335:p 300:Q 292:n 289:D 279:n 276:A 268:n 265:S 257:n 254:Z 31:. 20:)

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Index